AN ABSTRACT OF THE THESIS OF
PHILOSOPHY in
BINAYAK PRAASAD BHADRA for the degree of DOCTOR OF
Agricultural and Resource Economics presented on
Title:
June 2, 1981
SEPARABILITY TESTS ON WHEAT PRODUCTION FUNCTIONS IN OREGON
Redacted for Privacy
Abstract approved
/
John A. Edwards
Many natural resources such as water and forests have become more
intensively used in recent years.
Often, this has made it necessary
to reallocate these resources from less to more efficient productive
usage.
The knowledge of the existing tradeoffs between alternative
uses are necessary to make reallocative decisions.
However, these re-
sources also have strong public property character and are not usually
amendable to demand analysis to determine willingness to pay.
When
the 'price" is institutionally set, the productivfty measurement often
must be based on direct production function estimation.
For sectoral
reallocation of resources, some aggregate productivity measures are
required.
Such measurments are feasible when aggregate production
functions are estimated.
The aggregate production functions are however beset with a host
of difficulties arising from their aggregate nature.
aggregation bias must be eliminated if any
The resulting
gregate productivity mea-
sure is to be the basis of policy recommendation.
The improvement of
the methods
the results of aggregate productivity analysis hinges on
which reduce aggregation bias.
bias
There are two major conditions under which the aggregation
is minimized or eliminated altogether.
These conditions are a) rela-
and/or
prices are fixed amongst factors that are aggregated
tive
in an
b) the aggregated factors are weakly separable from others
economic sense.
The first condition relates to Hicks' Aggregation
Theorem and the second to Leontief Separability.
The latter condi-
tion appears to be directly relevant from the practical
standpoint,
since relative prices are seldom fixed amongst all aggregated factors.
aggregate
Thus the existence of valid aggregate input indices in an
production scheme can be assured only when there exists separability
between these inputs in each aggregate input indices.
The present study
producattempts to test for separability amongst inputs going into wheat
tion using county level Oregon Census of Agriculture data.
There is strong empirical evidence of weak separability amongst
husbandry prothe biological process inputs such as fertilizer and the
cess inputs such as capital and irrigation service.
weather variable,
And furthermore, a
rain, is found to be separable from both biological
and husbandry process inputs.
The tests were conducted utilizing the
functional
TRANSLOG type second order Taylor approximation to a general
form.
The separabilities imply various linear and nonlinear restrictions
and these reon the estimated coefficients of the translog function;
strictions were tested in conjunction with the usual F-distributed statistics of linear and non-linear restrictions on quadratic expressions.
The results from linear restrictions were however ambiguous between
capital and
inseparabilities among fertilizer and irrigation and among
irrigation.
Similarly, for nonlinear restrictions, ambigueity resulted
and
between inseparability amongst capital and fertilizer and capital
irrigation.
However, in both cases inseparability amongst capital and
squared
irrigation does marginally better in terms of the sum of the
error terms.
test for
A logarithmic cubic approximation function was used to
and husbandry
Sadan's perfect process complimentarity between biological
processes.
rejected.
This test using quadratic approximation models was
However, the strongest
evidence from the cubic approximation model
capital and irrigation
was that the model with inseparability amongst
complimentarity.
is the only one consistent with Sadan type high process
and husbandry
Thus the implication is that, at the micro-level biological
compliprocess functions are valid because of Sadan's perfect process
be dementarity, and,at the macro-level, wheat production function can
service, and ferfined for aggregate inputs of capital and irrigation
tilizer, precipitation, etc.
In conclusion, the results appear to be
biologisupportive of the lay notion that wheat production consists of
each
cal and the husbandry processes which are highly complementary to
other.
Separability Tests on Wheat Production Function
in Oregon
by
Binayak Prasad Bhadra
A THESIS
submitted to
Oregon State University
in partial fulfillment of
the requirements for the
degree of
Doctor of Philosophy
June, 1982
APPROVED:
Redacted for Privacy
rofesjr of Agricultural Economics
1/
in charge of major
Redacted for Privacy
ead of Department of Agricultural Economics
Redacted for Privacy
Dean of Graduate
Date thesis is presented
Thesis typed by
June 2, 1981
Mary Ann (Sadie) Airth
for Binayak Prasad Bhadra
ACKNOWLEDGMENT
I wish to express gratitude to Dr. John A. Edwards for his guidance
and constant encouragement throughout the course of this study; to Dr.
Roger G. Kraynick for his helpful suggestions and encouragement; to
Dr. William G. Brown for his helpful suggestions and encouragement;
to Dr. H.H. Stoevner, for his general guidance in the early part of
this effort.
I also thank Dr. David Faulkenberry and Dr. Ray Northam
for their helpful criticism.
I express my heartfelt gratitude to Centre for Economic Development and Administration, Nepal and Agricultural
Development Council,
for their continued support, without which the present research would
not be possible.
Sincere thanks are due to Mary Ann (Sadie) Airth for typing the
thesis.
I express my appreciation of moral support from my wife and daughter.
TABLE OF CONTENTS
Page
Chapter
I.
INTRODUCTION ...................................
The Problem Statement ......................
The Objectives of the Study ................
3
4
Background .................................
6
Various Approahces to Aggregation
Functional Form Approach ...............
8
8
Statistical Approach ....................
II.
III.
1
9
Process Function Approach ..............
Hicks' Aggregation Theorem .................
Leontief Separability Theorem ..............
Dynamics and Aggregation Over Time
Separability Theory ........................
Implications of Separability ...............
11
PROPOSED APPROACH ..............................
33
Aggregation Along the Processes ............
Reduction of Aggregation Bias ..............
The Process Functions and Separability
Reduction of Multicollinearity .............
Process-mix Optima and Sadan's Partial
Production Functions .....................
The General and Specific Hypothesis
33
34
METHODOLOGICAL BASIS ...........................
43
Theory of the Test of Separability .........
Flexible Functional Forms and TRANSLOG .....
Exact Translog and Translog Approximation
Other Forms and Monotonic Transformation
of Variables .............................
Econometric Estimation and Test of Hypotheses ...................................
Test Statistics for Linear Restrictions
Test Statistics for Non-linear Restrictions
Negative Random Error Model Under Sadan
Complimentarity ..........................
Sadan Model and a Simple Cubic Approximation...................................
Nested Hypothesis Sequence .................
Single and Multiple Partitions Separability.
13
19
23
25
29
35
36
37
41
43
45
46
48
49
51
53
56
60
63
66
1!?
MODEL SPECIFICATION AND HYPOTHESES ................
69
Model and Maintained Hypotheses ...............
Proposed Hypotheses ...........................
Regional Production Function ..................
Summary of Hypotheses ..........................
Sources of Data ...............................
Data Description ..............................
The Units of Variables ........................
69
71
76
78
79
80
84
85
V.
RESULTS...........................................
Linear Computational Procedures ...............
Model-A Results ...............................
Regional Difference in Production Functions:
Chow Test Results ...........................
Test for Cobb-Douglas Structure ...............
Test for Linear Homogeneity of TRANSLOG
in F, I, N, and K ...........................
Test of Linearly Restricted Weak Separability
Single Partitions .........................
Double Partitions .........................
Mitigation of Multicollinearity ...............
a) Models with Fixed K/N Ratio ............
b) Models with only KN Term ...............
c) Models with N or N2 and K or K2 Only
Nonlinear Computational Procedure .............
Quadratic, Cubic and Quartic Approximation
to Sadan Model ..............................
Quadratic Approximation to Sadan Model ........
Cubic and Quartic Approximations to Sadan
Model.......................................
Negative Error Models and Process Functions
Additive Error Model ..........................
Model-B Results ...............................
VI.
85
87
87
92
94
97
97
99
104
105
108
110
113
120
122
124
133
138
141
SUMMARY OF RESULTS ................................
145
Total and Marginal Productivities based on
the Cubic Approximation to Sadan's Perfect Process Complimentary Model ............
150
CONCLUSION ........................................
159
LIMITATION AND RECOMMENDATION .....................
162
BIBLIOGRAPHY
166
APPENDICES ........................................
172
AppendixA ....................................
AppendixB ....................................
172
180
182
183
185
187
Appendix C ....................................
AppendixD ....................................
Appendix E ....................................
Appendix F ....................................
LIST OF FIGURES
Page
Figure
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
a) Isoquants, b) Isocosts in f-g space ...............
a) Four Quadrant Analysis of Isocost Lines,
b) Optimal Process-mix ................................
Sadan Model: Perfect Process Complementarity .........
Perfect Process Complementarity .........
Sadan Model:
Perfect Process-complementarity and Discontinuous
Yield-function in Y-f-g space ......................
Discontinuous Yield-surface in Y-f-g space ............
Inverted Inclined Cylinder ............................
a) Isoquants, b) Inverted Inclined Cylinder ..........
a) Isoquants, b) Inverted Inclined Cylinder ..........
Nested Hypothesis Testing .............................
Iterative Non-linear Least Squares Algorithm ..........
b) Cubic Approximation
a) Eight Order Approximation,
Surfaces to Fit Scattered Points Along AB .............
Fertilizer Input, Marginal Product and Expected
Wheat Yield Based on Cubic Approximation
(P=lO in, K$2O.00) ...............................
Fertilizer Input, Marginal Product arid Expected
Wheat Yield Based on Cubic Approximation
(P=l0 in, K=$20.00) ...............................
Cross-section of Cubic Approximation Function
Yield vs. K/F Ratio ...............................
38
39
40
56
57
60
61
61
62
65
115
126
127
151
152
155
LIST OF TABLES
Page
Table
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
Chow-Test Results
Farrar-Glauber Analysis of Multicollinearity .........
Test for Cobb-Douglas Structure .....................
Test for Linear Homogeneity .........................
Single Partitions (with pairs) on Eastern Oregon
Model............................................
Single Partitions with Triplets on Eastern
Oregon Model .....................................
Double Partitions ..................................
Conditional Double Partitions ......................
Models without T and with Fixed K/N Ratio
Represented by K .................................
Model with Cross-product Term KN ...................
Models with Single Terms with N and K-variables
Non-linear Weak Separability Restrictions ..........
Inverted Cylinder (Quadratic Approximation to
Sadan Model) ......................................
Approximations to Sadan Model using KI Nonlinear
Term in the Process Function, g ..................
Approximation of Sadan Model using KI Term in
the Process Function, g ..........................
Approximation of Sadan Model using Fl and KI
Terms in the Process Function, g .................
Negative Error Models with KI Term in Husbandry
Process Function .................................
Ninety-five Percent Confidence Limits on the
Coefficients of Cubic Approximation Model
and Negative Error Process Function Models
Negative Error Models with Fl Term in
Husbandry Process Function .......................
Additive Error Model for Cubic Approximation to
Sadan Type Process Complimentarity ...............
Separability of Weather Variables P, T and Input
F,I or Separability Test of [P,T,(F,(K,I,N))]
88
90
93
94
97
98
100
103
107
109
111
119
123
125
128
131
134
135
137
140
144
LIST OF GRAPHS
Page
Graph
1.
2.
Cropland Harvested and Tractors (1974 Oregon
Counties) .........................................
KI vs. Fl, Eastern Oregon ..........................
149
Separability Tests on Wheat Production Function
in Oregon
I.
INTRODUCTION
The use of many natural resources such as water and forests have
expanded rapidly in recent years.
This has made it necessary to improve
their use efficiency in individual applications and also has raised
issues about reallocation of these natural resources from less efficient use to more efficient ones.
Many such reallocative decisions
have to be made both by the society and private entrepreneurs so that
efficient use of the limited natural resource is assured.
Such allo-
cative decisions can only be made on the basis of the tradeoffs that
exist between alternative uses of these resources.
Thus measurement
of the productivities of such natural resources as water and forests
in various uses have become important from the policy formulation point
of view.
The increasing conflicts of interest has made it necessary
to formulate these policies of resource use for the attainment of economic efficiency.
In case of many natural resources, traditional demand study involving price-quantity relationship have failed.
This is caused by the
public property nature of these natural resources.
For example, water
is allocated on the basis of traditional prior appropriation doctrine
in the Northwest U.S.A.
The
price" of the water is institutionally
set and does not reflect the willingness to pay on the part of the
users.
Thus without a market determined price, the demand relation
2
cannot be estimated to infer the marginal water productivity in agriculture.
The resource allocation is socially inefficient in such
instances and the policy formulation usually must fall back on measurement of resource productivities through direct estimation of the production function.
This is one reason for the recent emphasis on the
use of duality approach in applied production theory.
This approach
may be well suited to both the development and econometric application
of the production theory.
In production function studies, the multitude of activities con-
tamed within a sector often requires considerable simplification for
manageability of data gathering and subsequent analysis.
The majority
of the studies, at least in agriculture, appear to be either micro
response functions based upon experimental data or the aggregate macro
level sectoral (farm income) production functions based on aggregate
state or county level data.
This situation is expected since majority
of data is available at these micro and macro levels.
Though for
general policy formulation the macro level production functions are
relevant, these studies may incur extensive aggregation bias.
The
aggregation bias can result from aggregation of inputs, as well as
from the aggregation of outputs, when there are multiple outputs present.
Where all the outputs are produced individually in an indepen-
dent manner, the issues of aggregation bias can be reduced to the
issues of the bias of aggregating the inputs within a single output
production scheme.
3
The Problem Statement
Conceptually, a complex production scheme can be broken down into
simpler sets of activities, which are considerably fewer in number
than the list of all the activity inputs.
Thus, aggregation bias may
be thought of as a result of misspecifying activities in terms of the
wrong inputs.
In this case the solution to the problem of eliminating
the aggregation bias is to identify a set of inputs that truly belong
to each given activity or process.
This subdivision of the production scheme into processes requires
intimate knoiwedge of the production activities, and therefore of the
technology.
Further, such knowledge should be empirically verifiable
so that the existence of such processes and activities can be demonstrated.
The economic significance of separate activities and processes
lie in the fact that inputs into one activity can be altered without
affecting the other activities.
This is called the economic separa-
bility of the inputs in one group from those in the other.
The sim-
plification of a complex input-output relation can only take place
through the notion of processes and sub-processes.
If this conceptual
breakdown of the production scheme is not feasible, then one cannot
expect to be able to aggregate inputs without bias, and all the benefits of such simplified descriptions of the technology are lost.
Thus we can broadly define the problem posed in this study.
Given
a set of disaggregate input data and single output data, the problem
consists of finding the best way to aggregate the input data to minimize the aggregation bias in aggregate production function estimates.
4
Conceptually, this problem can be posed for any level of production
relation.
The production relation can relate to the macro-level as
well as the micro-level.
For example, if the data is available for
farm activities, the aggregation bias issues may be raised about the
aggregate inputs explaining the farm level gross-output.
Alternativ-
ely, if the data is available in disaggregate activity levels for the
inudstry, the best method to aggregate these input data to explain
the industry gross-output may be sought.
In the present context, the specific problem is posed as the
following question:
What is the method of aggregating the wheat pro-
duction inputs which generates no aggregation bias in the wheat production function.
If such a method is available, it could pave the
way towards an unbiased farm production function.
The Objectives of the Study
In view of the previous problem statement, the following objectives have been set for the present study.
(1) To test the separability of inputs going into the wheat production function in Oregon, with the view to establish the bias free
method of aggregation of inputs into aggregate inputs.
(2) To infer the validity of the notions of the processes within
the wheat crop-growing activities through the use of the notion of
economic separabilities of the process inputs.
(3) To test for complimentarities and substitutabilities between
the processes
(if they are defined), within the wheat growing activity
5
in Oregon.
In particular, tests for complementarities between biologi-
cal process of plant growth and non-biological husbandry processes
are to be devised, if these processes are found to be separable.
(4)
To infer the productivities of the factor inputs from the
estimated production function, and the aggreçjate input functions.
It may be emphasized that, the purpose of the present study is
merely to test the validity of aggregation, as is usually performed
in aggregate production function studies taking the wheat production
function as an example.
The present study is not intended to carry
on with the extension of separability tests for other crops and agricultural activities with a view to 'construct" aggregate farm-output
production function without aggregation bias at the macro-level.
Though
such an attempt would reward one with better estimates of sector level
factor productivities, which could be of immediate policy relevance,
it is beyond the scope of the present study.
Background
The theory of production, which deals with the decision making
process of a producer unit (maximizing profit for a given level of
resource endowment) treats products and factors as well defined entities.
However, when we look at the empirical application of the theory,
we find that the 'products' and 'factors' are actually aggregates of
distinct goods and services.
We may thus question whether or not
the theory has any empirical relevance.
Otherwise, there must exist
conditions under which use of aggregate factors and products are justified in production theory.
Further, these conditions must be testable.
It is often noted that the variables used in production and derived
demand studies are invariably some kind of aggregates.
gation is ever present, it is sometimes argued that
Since aggre-
there cannot be
an empirically meaningful way of dealing with aggregation bias.
In
what follows, this will be shown to be false.
The study of factor demand and factor productivity has been traditionally conducted using aggregate and disaggregate (experimental)
production functions [Ruttan (1956), Heady (1957), Hock (1962),
Holloway (1972), Thomas 1974), Lynne (1978), and Mitteihammer et al.
(1980)].
Factor demands are estimated using aggregate demand
[Cromarty (1959), Griliches (1959), Heady and Yeh (1959), Kako (1978)].
7
The latter approach can not be used for inputs without an observable
market price, such as water.
Lynn (1979) has indicated that non-
market and the public property nature of water is primarily responsible for this.
This indicates that, for many common property type
resources, derived demand estimation approach fails.
Thus resource
use policies must be based on direct production function estimation
and direct factor productivity measurements in these cases.
Thus the
issue of aggregation bias is a pertinent one in these aggregate production function studies.
[.1
L.]
Various Approaches to Aggregation
There have been three distinct approahces to deal with the issue
of aggregation in production and factor demand studies.
The usual
approach has been to assume a true functional form for a production
function and then to proceed from there to determine the best way to
aggregate data.
The main concern of course is the extent of bias that
is generated due to aggregation.
In contrast to this functional form approach, there are approaches where the knowledge of the true functional form is not presumed.
The functional form in these approaches is empirically decided.
The
pure statistical aoproach and the Process Function AoDroach fall into
this cateqory.
(i)
These will be discussed separately below.
Functional Form Approach
For a log-linear form of production function, Klein [1946] was
the first to show that, the best way to aggregate independent variables
was to take their geometric means rather than the arithmetic ones.
He also indicated that if the arithmetic means are used instead of
the proper geometric means, a downward bias occurs in the estimated
coefficients of the log-linear form.
The reason is that for small
variations the geometric mean G can be shown to be approximately equal
to X [1 -
12
(2)] < 'X, where, X and
are the arithmetic mean and
x
the variance of X, respectively.
Nataf (1950) showed that for a sensible aggregation of the micro
function to a macro one, the assumption of additive (in factors)
I!]
production function is necessary.
Thus, because Cobb-Douglas produc-
tion function is additive in log-linear form, use of geometric means
mostly eliminates aggregation bias in estimation of the aggregate production function.
For a log-linear form, a simple average of the
logarithms of the variables turns out to be the logarithm of the
geometric mean.
The geometric mean, G =
X1)
e
= e
n X
Xn) can also be expressed as,
f(x1x2
Thus,
n G =
model, the relevant average is,
ate method of aggregation of the
n X1 and for the log-linear
.
n X..
n X
Thus the appropri.-
log-linear form of micro-level func-
tion to a log-linear form of the macro-level function is to use the
geometric averages instead of the arithmetic ones.
Similar methods
of aggregation of variables into "means" could also be derived for
other functional forms (e.g.,the Constant Elasticity of Substitution
form) which can be changed into linear functions through transformation of the variables.
(ii) Statistical Approach
Another approach that has been indicated to be feasible is the
statistical approach.
Theil (1954) poses the question as to the pos-
sibility of fitting a macro-relation to the aggregate values when the
micro-relations are given along with the form of aggregation [Walters
(1963), p. 10].
For example, let the micro relations be Cobb-Douglas
production functions in their logarithmic form, for each
X
where, X.., i, k
lk1 + a
i
th
fj
= 1,..., n.
represent output, labor and capital, respectively in
10
logarithms.
Then we may attempt to estimate a macro form,
X =o
1 ±3k
+ a
using the aggregate values of a time series.
The individual firm demand for inputs could be estimated by the
following regressions,
ii =
k
1
ii
+
Ck
+
+
= Bkj 1 + Ckjk + 0ki + Ukj
utilizing the aggregate variables 1 and k.
The regression coefficients
D, B and C represent the connection between the macro and micro var-.
iables.
They represent how micro quantities have to move when the macro
Thus, knowing 0, B and Cs we can estimate a macro
variables move.
function, X =ol +3k +
a +
firm production functions:
,
by substituting for ii and ki in the
X. =ol +3k
+ a:
for all is.
The
macro function parameters [Walters (1963)] are,
=
+
n
Coy
(8 Bi)
+ Coy
(3 B1)
3
=
+ n
Coy
(
Ci)
+ Coy
(3 jCkj)
a
= n
+ Coy
(3 .Dki)
+ Coy (
i0li
One advantage of this approach is that it gives the covariance terms
above as the aggregation bias in the macro parameters, oi.
ever, not knowing a., 3,
covariance terms a priori.
, 3 ,
a.
How-
it is hard to estimate the size of these
This appears to be a major drawback of
Thei1s approach from the empirical standpoint.
Houthakker
(1955) has also shown that if the input/output ratios
of the firm level linear production functions have Pareto type
distribution the aggregate production function for the industry as a
11
whole is in the form of a Cobb-Douglas.
Though this represents an
adequate approach to cross-sectional data, there still remain questions
when the input/output ratios are non-Pareto distributed.
Thus in prac-
tical terms, this approach is also inadequate.
(iii)
Process Function Approach
Another approach to aggregation is based on aggregating the process
functions.
These functions are defined according to the analytical
conveniance of the scientist or the engineer [Chenery (1949); Ferguson
(1953); Heady and Dillon (1962)].
They have been estimated on the
basis of agronomic or engineering data [Moore (1959); Markowitch (1953);
Hirsch (1952 and 1956)].
functions to
'factory1
Attempts have been made to integrate process
or 'plant' functions or even 'firm' level pro-
duction functions [Smith (1961)].
Problems with the process functions
have been that they normally exclude indirect inputs such as management
machinary, land, etc. from consideration [Smith (1961)].
Process and
'plant' production functions are useful in deriving the 'plant' cost
curves, though they are not useful for asking questions about the returns to scale at the farm or 'plant' levels [Walters (1963)].
When process functions are used for the farm as a whole, specification bias results from ignoring the indirect inputs (management)
into the farming process [Grileches (1957); Mundalak (1961)].
Comparative Statics and Aggregation Bias
The use of aggregate variables in production and derived demand
studies can be justified in three fundamentally different ways.
The
12
simplest way is to use the notion of a linear technology, such that,
the input-output ratios within aggregated inputs are fixed.
This kind
of perfect complementarity within a subset or inputs is unrealistic.
Therefore the assumption of linear technology is not a generally acceptable rationale for the use of aggregate inputs in empirical studies.
A better justification, popularly given, is based on the important
theorem of comparative statics, known as the Hicks' Aggregation Theorem.
Unfortunately, the theorem holds only for the unlikely situation where
the relative prices of the aggregated inputs do not change at all.
The final justification for aggregation is provided by the existence
of a restricted type of technology, characterised as weak separability
of aggregated inputs.
Though all the three justifications above are testable empirically, this has generally not been done.
So, as will be argued below,
there is always a possibility of excessive aggregation bias in studies
which utilize aggregate input and output data.
More importantly, how-
ever, the three justifications can also furnish some insights into
the style of aggregation which will result in minimum aggregation bias.
This is an important consideration wherever there exist disaggregate
data requiring aggregation for manageability and ease of analysis.
In the next sections we will deal with the Hicks' Aggregation Theorem
and Leontief Separability Theorem, respectively.
13
Hicks' Aggregation Theorem
In Value and Capital (1946, pp. 311-313) Hicks demonstrated the
following very important aggregation theorem.
The theorem states that
if the relative prices within a group of commodities are fixed, the
value aggregate of such commodities behaves exactly as if it were a
separate intrinsic commodity.
This is the basis of using Hicks-Allen
money in popular two dimensional geometry of demand analysis.
If the prices p1.
.
p
of the commodities X1, ..., Xr move
in exact proportion, we can define the composite commodity, as,
r
with the usual properties of a single commodity.
X1 =
Samuelson warns that this definition merely results in the change of
frame of reference and not in any change in the dimensionality of the
problem from n to (n-r+1) (see pp. 142, The Foundation of Economic
Analysis).
Samuelson has originally shown (The Foundations of Economic Analysis, pp. 129-143, 1947) that the comparative static results remain
invariant under some kinds of transformation of variables, provided
these transformations preserve value magnitudes.
Let the price and
quantity vectors p,X, respectively be transformed to new coordinates
,X such that p' X
'X.
This has been called a contragradient linear
transformation of p,X into j5,X by Samuelson.
He shows that we can
indicate the transforntion through a non-singular matrix c, such that,
=
c1x,
= c'p (Note p
and
'
are transposed p and
respectively).
Consider the production problem, where output y=f(X) is to be
maximized under an expenditure constraint M=pX.
The Lagrangian
14
associated with this problem is, max L=f(X) + X
(M-ptX), where,
x,x
Under the contragradient linear transfor-
X = Lagragiari multiplier.
mations, the Lagrangian is, L=f(c) + x (M-'), the form of which
Thus we can re-state the problem by the Lagrangian.
has not changed.
max
x,x
() = f(c)
+ x (M-) where
=
The necessary conditions for the maximum of L(X,A) are,
= 0
or
f
- Xp
= 0
or
M
-
= 0
1=1,... ,n
n
E p.X. = 0
i=1,.. .,n
ill
and the sufficient condition is that the bordered Hessian,
2
I
[x1
p
x
-I -
T32L
-
-
0
L
is a negative definite matrix of size (n+l) x (n+l)
Analogously the necessary and the sufficient conditions for the
maximum of 1
(X,x), in the new coordinates K, are
= 0
or
- X
.
1
= 0
i=1,.
.
. ,n
1
xi
n
or M =
0
x.
x.
arid
T =
-
1
-
= 0 i=1,... ,n
ax..
_1_
x
x.
_1_
x.
3j_
-
I
0
2X2
15
But it should be noted that, the negative de-
is negative definite.
is also negative definite, and there-
finiteness of T implies that
,is possible and
fore the optimization of t using transformed inputs
val id.
Further, explicit differentiation of,
1M=
gives the following matrix equation,
[1
x.
I
x.
1
where, X
and
rI
=
P
Define,
.
FJ_
and p = f
fiJ [Xj 1:3p1
LP
[X
[J
[c' ol I T_1J{c
0
0
1'
iJ
= B
B
and B' represents a congru-
ent transformation, it preserves the definitness of
negative definite.
But,
i1=
X
T4,
T1.
Therefore,
we therefore conclude
-
r
that, T =
J so that
3[MJ
(g
Because multiplication by B
[iJis
AT'
p1
and
[J= r-1
or
=
_0_ j
I
Thus,
p4X4
1
2f
3X
{ x
i=1,... ,n
=
[i
KJ;
is also a negative definite matrix.
There-
I
fore, [-.J matrix is embedded with the same qual ities as
matrix.
Therefore, the following relations hold for the transformed input
variables, L analogous to the untransformed X.
I1
The
Slutsky equation,
(ax1
\a p
/y constant
IThi.I
where, __Lj
=
= k.., is
JT
Iy=constant
th
explained as the, residual variability of the
th
variable for a compensated change in the
th
It is the pure
(-) represents the expenditure
Similarly,
or budget effect of
price.
transformed input variable, X, due to
substitution effect for
change in price of
transformed input
price change.
The negative definite T implies the inequalities characteristic
of conditional factor demand, 3
e.g.
i
(,y), e.g.
=
a
= k.. < 0, and k11
22
1J
- k12 > 0, etc.
alternate in sign.
so that3the principle minors of 1
Thus the demand
= K(,y) satisfy the same inequalities satisfied by Un-
functions,
transformed variable demand functions, X = X (p,y).
Now let us define the contragradient transformation on X so that
r
the first element of X, X1 = E
is the composite input
i=i
X1
1..2::Pr0::0_7
i
=
X
0
xn
ri
-
j
'n-i
Ii/p11
C'.
X
0
1
11
P2?Pit
:
Cp.
I
p
L
J
'n-i
0
lr+1j
¶
0
I:
jLPnL
0
J
L
x
17
Because, p'X =
the value magnitudes are preserved as required
for this particular, c.
1'"'r
It should be noted that when the prices
with respect to
change in the same proportion
changes from 1.0 to Q.
all held constant, only the first element of
The other (r-1) elements of p,
r+1'" ,
all remain fixed at
'r
all remain fixed at zero, and
r+1''n'
respectively.
Thus under the new coordinate transformation, the proportional
change of first r prices p1,...
'r
is transformed to a single change
price of the composite input commodity,
in
However, we have
,
behave
already shown that the composite input variables, Xi,, X2,.
exactly as the original variables X1,... ,X
optimizing behavior.
under the assumption of
The same necessary and sufficient conditions
were derived for contragradient transformed variables, X as is true
for X.
Because negative definite T implies negative definite T, we
can show that there are qualitatively the same properties for the factor demand functions, X = X(p,y) and X =
(,y).
It should also be
noted that the Lagrangian multiplier is independent of the transfor-.
mation, so that, A
= X (p,y) =
(,y).
Thus we conclude that, under the condition when the relative prices
of Xi*Xr are fixed, contragradient transformation on X1,..
which
may be performed to define a composite commodity,
has all the economic attributes of a single commodity.
This is Hicks'
Aggregation Theorem in its simple form for a single group.
It can
be extended to more groups.
In conclusion, as long as the relative prices do not change within a group, the production function, y
(X), is a valid production
18
The production
function in terms of the composite input variables, X.
function, f(X), under prices,
constraint, M=', and the maximizing
,
behavior, generates, the usual conditional factor demand function,
Note,
=
=
'°'r+1''n'
Thus, factor demand can be analyzed with (n-r+l)
variables using the X(.) relations.
This aggregation theorem, though proven briefly
Samuelson, is proven precisely by Gorman (1953).
by Hicks and
Though the result
of
the theorem is strictly valid only for fixed relative prices within the
aggregate inputs, Diewert (1978) has shown that even if the individual
prices do not change in exact proportions, Hicks' Aggregation Theorem
holds approximately.
Though this allows for small changes in relative
prices, it.is difficult to judge the quality of approximation at hand.
More importantly however, the relative prices can be stable only
under specialized situations over a long period of time.
One possi-
bility is that the inputs are close substitutes, so that if the price
of one changes the price of the other also changes in the market.
On
the other hand, we can also have the extreme opposite, i.e. perfect
complementarity.
movement.
The latter would also result in close relative price
The importance of the theorem is undermined because both
these requirements are often not met in reality.
Thus if the price changes in the aggregated group are not in the
same proportion, the use of aggregate variables in production function
estimation results in an aggregation bias.
It is expected that the
bias would increase with the increase in differences in relative prices.
In reality relative prices don't usually stay the same for a long time
19
and aggregation bias would result if the Hicksian theorem alone provides the basis of aggregation.
If the aggregation bias could be fore-
told a priori, the changing relative prices would not be detrimental
to aggregation; compensations could be provided to correct the bias.
Unfortunately this is not so.
So we now turn to Leontief Separability
Theorem.
Leontief Separability Theorem
The final justification of aggregation is provided by the property of weak separability of the overall production function.
Though
this notion of separability is mathematical in character, its origin
Leontief originated the concept of separability
lies in economics.
The 1947 article in Econo-
in the context of a production function.
metrica,
Introduction to a Theory of the Internal Structure of Func-
tional Relationshipsu, starts out with the problem of simplifying an
over-all production function Y = F (X1,.
into a function of
r < n.
few
a
.
.
,X) of variables X1,...
intermediate variables f1,... 'yr' where
These intermediate variables, f1 's are themselves functions
of subsets of the variables X1,... ,X, such that the subsets are
So the following identfty would hold,
mutually exclusive.
A.
(
(V
1
V1
,...A
\
(v
c
\A2
r
yr
r
According to Leontief, a complex production scheme can be represented through a set of intermediate production functions,
f,(X1,...,X'1
I
),
(See pp. 362, Econometrica, 1947) provided there
1.
exist appropriate technical informationson the intermediate steps of
the overall production process.
Though the knowledge of the
20
intermediate production functions f1(..) can be easily combined to
construct the overall function,
(f1
,.. 'r'
the reverse process
given the properties of F(X1,.. . ,X) is not
of determining f1,..
easy.
Leontief thus raised the question about the conditions
of F(..)
or nature
under which the over-all production function, F (..), can
be functionally separated into intermediate production functions, as,
(f1,..
F C..)
This pertains directly to the justification
'r
of aggregation in production and demand functions.
th(fifr) we are justified in aggregating inputs
expressed as
X1,... ,X
If F (.) can be
into r mutually exclusive categories, each represented as,
1'"' nF
In another 1947 article, A note on the Interrelation of Subsets
of Independent variables of a Continuous function with Continuous First
of the American Mathematical Society), Leontief
Derivaties (The Bulletin
demonstrates that the necessary and sufficient condition for weak
separability of input set X into the mutually exclusive complimentary
subsets S and
is that the "rate of technical substitution" between
two inputs X., X. in S is independent of input Xk in S i.e.
1
F/X.
=
k BF/X
where F(X)
(f(s), g()).
The necessary part of the above result can be easily demonstrated
as follows:
-
B_o
-
f
.
1,3
F/X
and therefore,
-ç (F/x)
f/X
0.
21
F/aX.
x1] = 0, is also sufficient for, F (.)
This condition,
Xk
'
This can be shown as follows:
Define,
'
(S) = F (S,0), where
represents the set of all Xk fixed at the value, X.
S consisting of X1,... ,X
o ( ti1(S),).
Now, the subset
is locally functionally separable in X if the
function F(X) can be shown to be expressible in
F(X) =
(f,g).
j
'(S) and
,
i.e.
Let us now consider XkS in S as parameters of
F(X), then we can construct the following matrix of derivatives of the
function F(X) and '(S) with respect to X1,...,X,,.
....
The first row of the matrix above is proportional to the second
row if, we have,
F/X.
1
1
for i,j < v.
F/BX
Iu/BX
This is true when, Xk = X
for all
k > v.
F/X.
= 0, for i,j < v and k > v, we must have,
Because, -i-
F(S,0)/X1
'Y'/Xj
as the ratio is independent of all
Xk s.
Therefore we see that the rows of the matrix are linearly de-
pendent on each other.
However if we make the usual assumption that,
> 0, then according to the general theorem on functional dependence
we can express, F(X) =
(F(S,0),)
=
could be repeated by taking the first set
etc.
Then we would conclude that, F(X) =
(y (s)J).
The same arguments
and defining F(,S°) =
22
Thus, in general, F(X) =
F(X) =
(' (s), V(')) which we recognize as,
q(f,g).
We recognize that Separability is a very restrictive assumption.
Berndt and Christensen (1973) have shown that Leontief's Separability
Conditions imply that partial elasticities of substitutions between
input X
i
in S and input X
< v and k > v.
in
must remain the same locally for all
This result will be demonstrated later when we deal
with Separability Theory.
The implication of this result for the usual aggregate production
function in agriculture is devastating.
Consider, the input category
expenditure in the usual Cobb-Douglas farm production function.
Expen-
diture is an aggregate of fuel, fertilizer chemicals, pesticides, etc.
Now, because Cobb-Douglas is log-linear, it is also weakly separable in
all inputs, capital, labor, etc., including expenditure.
The Berndt-
Christensen results thus imply that, fuel and capital have the same
elasticity of substitution as fertilizer
chemicals and capital.
Realistically, fuel is complementary to capital and a substitute to
fertilizer.
Thus the usual pattern of aggregation of fuel, fertilizer
chemicals into expenditure group appears highly tenuous.
At this stage, it serves well to recognize that the
Hicksian Ag-
gregation Theorem and the Leontief Separability Theorem provide the
'necessary" basis for aggregation.
These considerations though para-
mount in importance, do not exhaust all.
Separability restricts the
functional form and so the aggregation issues are also related to the
choice of functional form to be employed empirically.
The functional
form, most appropriate at a given level of analysis (i.e. firm level,
23
sector level, process level, etc.), is necessarily consistent with
forms at higher and lower levels of analysis.
For example, the micro level process function may be presumed
to have increasing returns to scale, operate under limited entrepreneural ability, and often work under rather elastic factor prices.
For the macro-functions, constant or decreasing returns to scale, with
variable entrepreneural ability but faced with inelastic factor
supplies becomes a more realistic assumption.
So exactly the same
parametric specification of macro and micro-production function would
not be desirable.
But when micro level functional forms are known,
the criterion of choice of the macro function is that,
the model
give rise to a sensible aggregate relationship, which, in some sense,
corresponds to the micro-relations' [Walters (1963)].
The present analysis has so far preceded without considering the
issues of aggregation over time.
It can be demonstrated that pro-
vided the inputs are labeled separately for each time period, the
Leontief Separability Theorem can be easily extended to deal with aggregation over time.
In contrast to the comparative statics case
however, the presence of dynamic growth complicates the matter considerably under the dynamics.
So presently we will consider
aggregation over time by extending the static production function to
a dynamic one, the production functionals.
Dynamics and Aggregation over time
Under dynamics, the micro-level production relation should describe the process of growth and change at the micro level
.
For
24
agricultural processes, the process description of biological growth
requires Functionals rather than Functions [Georgescu-Roegen (1971),
The Analytical Representation of Process and the Economics of Production, Chapter 9, pp. 234-238].
The 'inputs
to plants and animals
are time functions (fund-flow rates in Georgescu-Roegen's definition)
and 'outputs' are resultant growths of bia-mass, again time functions.
This picture is immediately relevant to the case of crop and livestock
activities.
The dynamic nature of crop production has been well recognized.
For example, a few studies have asserted the need to explicitly con-
sider water application regimes rather than the aggregate water input
in crop production functions [Edwards (1963), Minhas, Parikh and
Srinivasan (1974), Stegman (1980)].
However, Edwards explicitly con-
siders the growth of plants in determining their yields.
In his
study, temperature-moisture regimes during various growth stages and
the previous growth is used to explain the final yields.
Though it appears plausible in many instances, that biological
growth achieved finally becomes a simple function of time aggregates
of inputs, this need not generally be true.
This puts a serious limi-
tation on the time aggregated production function as an approximation
to the dynamic functional.
Yet when the growth produces a final
cumulative result (say the production of a seed after plant matures),
the effect of all input flows may be assumed to be accumulated at the
end.
If the inputs at any stage of growth were independent of the
inputs required at a different stage, then this time wise separability
would allow,time-aggregated inputs in specifying an aggregate yield
25
function.
This independence of dated inputs however is quite contrary
to the known laws of biological growth.
requirements of plants and animals depend
The nutritional (and basic)
upon the present ambiance
and on the previously attained growth.
Further, even assuming the stage wise separability of inputs,
one has to consider that inputs required for
controls will always be
independent of controlled inputs at all stages of growth.
This may
not be a realistic assumption in general, and time aggregation will
not be generally valid.
If, however, these separability assumptions
hold, then aggregate yield functions with time aggregate inputs will
be valid, especially when the input applications cannot be reversed,
and
therefore become asymmetrically independent over time.
Separability Theory
In his 'Theory of Internal Structure of Functional Forms'
in Econometrica (1947), Leontief demonstrated that weak separability
was a necessary condition for the existence of a subaggregate index for
a group of inputs.
That is, a production funtion, Y = F(X1...X)
can be written as,
=
when the input set
can be divided into mutually exclusive
X1,...
groups or subsets X1 =
X,
..., X
for i = 1,2,.. .r, such that the
subsets are weakly separable from each other.
separability was shown to be
/
3Xk
F fF
3Xf/
-o
The condition of weak
26
Xr
when
1i\
r.
Alternatively,
FiFik = 0, also indicates weak separability, where
FjFjk
F.
X5, where, s
and Xk
32F
and F. = ----, etc.
OA
1
0A1 Ak
=
This last expression indicates that the marginal rate of substi-
tution (F/3X1)/(3F/X)
between inputs Xi and
independent of inputs outside this group, Xk.
.
fr(Xr
X
in group r is
Solow (1955) has called
a consistent aggregate index of inputs
)
X,
.
.
So a consistent aggregate index of a subset of inputs exists, if and
only if, the subset is weakly separable from all other inputs [Green
(1964)].
Berndt and Christensen (1973) showed that for homothetic production function's,the separability restrictions imply certain equality
restrictions on the Allen-partial-elasticity--of-substitutions (AES).
The AES is defined between two inputs X
1FiXz1ik1/ XiXkJ
ik
This AESik measures the response of derived
bordered Hessian of F(-).
demand of input X.
column element of the inverted
row and k
where, Hik is the i
and Xk, as,
to a change in price of input Xk, holding output
and other prices fixed.
Under the assumption of efficient production
and perfect supply elasticity of inputs, the AESik has the following
property,
=
1
3X.
nk
-iwhere,
ik
k
= PJXj/
dk
-, price of elasticity of demand, and
i
cost-share of Xand Pis the price of factor Xi.
For homothetic production functions, Shephard (1953) showed that
there exists a dual cost function of the form,
C (Y,P1,
g
27
where, e and V are some functions such that the cost is an increasing
The above implies that, if the production
function in Y and P's.
function is weakly (strongly) separable in X's then the cost function
would also be weakly (strongly) separable in the P's.
This was shown
by Shephard (1970) for separable homothetic production functions,
though Lau. (1969) had shown a similar result of homothetic separabi-
lity between direct utility function U (X1,
direct utility function, o (I)
'
(P1,
...,
X) and its dual
in-
P) [obtained by replacing
the optimal values of X..'s in U (X1, ..., XN) with their demand
expressions,
I
is the budget constraint].
This result allows an im-
Under homothetic separability, a consistent sub-
portant conclusion.
.
aggregate price index,
exists, if and only if, the
corresponding consistent sub-aggregate quantity index fr(X
...,
exists [Berndt and Christensen (1973, pp. 405)].
Under efficient production, and perfect factor supply elasticies,
Samuelson (1948, pp. 61-69) has shown the comparative statics result,
DX.
H.
=
for all k,
i
= 1,2, ..., N,
where X is the Lagrangian multiplier in the con-
and also F. =
strained optimization problem.
Substituting these into the expression for AES we obtain,
DX.
DX.
E
1
ik
for all i,j
3
Gjk
XjXk
DXk
XjXk
1, ..., N and E =
1
At this stage Hotelling's Lemma (1934) can be invoked.
This Lemma
states that, under the maximization hypothesis, the optimal factor
levels are expressible as,
(C(Y,P)\
3PJ= ()()
xi*
= e(Y).
i
=
where, C(Y,P) represents the cost function dual to the production function.
So that, we have,
X' = e(Y)
and
Also, E =
°k
PX
'i'd,
= 0(Y)
x
= e(Y)
etc.
'ik
C*(Y,P)
*(pp)
By substituting
where, * represents the optimal value of the variable.
we may show that
k
AESIk
and AESik
So that, AESik = AESik is true, if and only i',
ijk
Thus we conclude that the cost function C(Y,P)
@(Y)
is weakly separable in prices
AESik = AESJk.
from
jik
.
F (P1,
= 0.
..
,P)
if and only if,
k'
Therefore, weak separability of prices in the cost
function has been shown possible, if and only if, the homothetic production function is also weakly separable in the corresponding inputs
quantities.
Now we can conclude the Berndt and Christensen result,
that equality of AESik to AESJk implies that the production function is
weakly separable into the factor groups
(X1,
...,
XNr)
and
S
X1, ..., X) and vice versa.
The above result is derived for homothetic production functions.
Using the Shepard-Lau duality result (that homothetic separability on
the production side gives rise to a smiliar separability on the cost side,
in terms of corresponding prices of factors and vice-versa).
result is valid for the local point under consideration.
This
29
These results have been extended by Russell (1975) to nonhomothetic production functions.
It should however be noted that,
for Russell s general results to hold, the aggregator or separable
parts of the function,
thetic.
fr
(X,
. ..,
X) themselves need to be homo-
Russell s results also hold globally rather than locally,
provided some regularity conditions hold on the separable production
function.
The regularity conditions are that both the aggregate
function and the separable parts of that function be continuous and
at least twice differentiable.
Implications of Separability
Separability puts severe restrictions on the nature of technology and the form of the production function.
Since separability is
equivalent to equality of AES restriction, the following implications
are summarized:
(1) An aggregate net output or value added functions can be defined under separability.
(2) There exist aggregate input indices or separable parts to
the production function, (these will be referred tc as process functions
later on), and,
(3) There exists the possibility of having decentralization and/or
multistage optimization procedures.
For example, Bruno (1978) has shown that if the intermediate inputs are separable from direct inputs, the net output (or value added)
production function can be defined in terms of the primary inputs.
This is true even if the underlying production function with all inputs
30
is not homothetic or linearly homogeneous.
Denny and May (1978) have
tested the validity of separability in Canadian Manufacturing, and found
negative results.
This implies that unless the Hicksian aggregation
conditions hold, no real value added aggregate function can be defined
in this case.
Thus separability test may be used to establish the
validity of the empirically determined net value added aggregate production function.
Berndt and Christensen (1973) and Denny and Fuss (1977) tested for
the existence of consistent aggregates for labor and capital (using
U.S. manufacturing data) using the separability test.
dence for labor separability,
There was no evi-
but they found support for capital
aggregation provided labor aggregation is presumed.
of success in using separability
This is an example
to test for validity of aggregate
indices and functions.
Under separability, production efficiency can be achieved by Sequential optimization.
In consumer theory Strotz (1959), Gorman (1959)
and Green (1964) have indicated that strong separability allows a
budgeting procedure consistent with two stage maximization.
Blackorby
et al. (1970) showed that this budgeting procedure is also consisteit
with two-stage optimization, if and only if, weak separability holds.
When separability can be asc.ertained, the derived demand by fac-
tors can be simplified. Pollak
(1969 and 1971) has shown that the
factor demand functions are a function of the factor prices within
the separable group and the cost allotted to that group.
This does
not mean that the quantities of factors demanded in one separable group
are independent of factor prices in other groups or of total cost of
31
inputs.
All this means is that these variables enter into the demand
functions only through their effect upon the cost allotment to that
group.
The cost allocated to a group of factor inputs are frequently
known, so that the factor prices outside of the separable group can
be ignored altogether.
This results in a considerably simpler demand
equation to empirically estimate.
When factor demands are so simplified, the dual cost function
will also be weakly separable in corresponding factor prices [Shephard
(1970)].
Separability also opens up the possibility of multistage
estimation of macro production functions using consistent aggregates
in later stages.
For example, Fuss (1977) has used a two-stage pro-
cedure to estimate demand for energy in Canadian manufacturing.
The
net output or value added function can be legitimately estimated without considering the intermediate inputs, and Bruno (1978) shows that
the marginal productivity of the primary inputs will be estimated
accurately without bias.
Agricultural value added or farm income
only be legitimate under separability.
aggregate' model would
The confirmation of the
separability assumption in empirical terms is necessary prior to
deriving any policy recommendations based upon possibly biased, aggregate value added functions.
Under separability, there exist aggregate input indices or separable parts to the production function.
These indices or separable
parts of the production function will be called process-functions later
on.
The reason for calling these separable parts "process-functions"
32
is as follows.
All the inputs that belong to a separable part are
independent of inputs in other separable parts.
Thus, these inputs
within a separable part directly interact only with one another.
Therefore, these direct interactions may be summarised as a
'process.'
In the following chapter, it will be argued that various lay notions
of "processes" or "activities" may coincide with the separable parts
of the production function.
Thus, the test of separability may be used
to provide empirical support for the existence of process functions.
If separability assumption holds, then it provides a natural basis
for aggregation of inputs in a production function without introducing any aggregation bias.
This approach of aggregation along the
process functions will be dealt with in the following chapter.
33
II.
PROPOSED APPROACH
Proposed Approach:
Aggregation Along the Processes
In agriculture, there exists substantial heterogeneity across
farms at the aggregate level.
The farms differ from one another in
terms of the types of activities pursued and also in terms of their
intensities.
It is conceivable that a large portion of the observed
heterogeneity results from the variation in the mix
of similar
farming activities or processes.
The underlying biological processes within each activity is
however homogeneous.
which involve
The same is true of
most husbandry processes
tillage, seedbed preparation and
for similar crops).
harvesting (especially
Thus a complex farm production scheme can be
divided into simpler homogenous subprocesses.
The availability
of process-wise data provides the basis for testing this homogeneity
within mechanically and biologically similar processes.
This inherent integrity of the underlying biological processes,
and to some extent, the mechanical processes, also implies that these
farming subactivities
economically separable.
are independent of one another, and therefore,
It is noted that a farm may exploit compli-
mentarities between these otherwise independent processes.
This
separability between inputs going to different activities indicates
the route through which aggregation bias may be minimized in aggregate
production function estimation.
The underlying heterogeneity of
34
farms themselves indicates that farm wise aggregation of inputs
and outputs is undesirable.
The separability of various activity inuuts validate an aggregate production function of aggregate inputs, where factors are
aggregated along the activities and not across farms.
Reduction of Aggregation Bias
The aggregation bias would be minimized if aggregation occurs
along homogeneous processes rather than along heterogenous farms.
If we limit ourselves to a cropping activity, and if all the
mechanical and biological processes are spatially replicable (i.e.
linear homogeneity) and similar factor ratios prevail, the aggregation bias will not increase in increasing the level of aggregation
from one farm to a group of farms.
Though factor ratios vary
within the same process, they may vary very little.
This
indicates the possibility of utilizing the county level data
available from the Census of Agriculture in the U.S.A.
For a single crop growing activity, the subactivities of
tillage, seedbed preparation, fertilization, irrigation and harvesting
are quite homogeneous within and between farms.
The activities of
extension agents and the standardization of machinery also contribute
towards increasing this homogeneity.
However,
in spite of the
highly plausible nature of process homogeneity and separability,
it is at the end, an empirical issue.
Thus for the validity of
an aggregate crop function, the separability of these subprocesses
35
must be demonstrated.
Separability of inputs into these subprocesses
must be tested before attempting an aggregate crop production
function.
For example, for wheat crop, the husbandry process of tilling
and havesting requires fuel, capital and labor.
On the other hand,
the biological process of plant growth does not require these inputs
after seedbeds are prepared.
Plant growth does, however, require
moisture, sunshine, fertilizer, pesticides and herbicides, etc.
Thus, based on the sequential nature of the husbandry and biological
processes, we may deduce that the fuel, capital, labor inputs
are independent and weakly separable from fertilizer, moisture,
pesticide and herbicide inputs.
The Process Functions and Separability
When a production function is separable in inputs, and the
inputs in each group are related to a specific subprocess,
the separable part of the production function can be termed the
process function.
When the output of the process can be determined,
the process function can be estimated empirically.
An example
of a biological process function would be a response function.
These have traditionally been estimated to determine factor productivity and substitutability under a given experimental set
up [Moore (1961), Heady et al. (1956), and Berringer (1961)].
Some studies seek out the effects of random weather variables
[Edwards (1963), Hillel and Guron (1973)] and others that of
ri
changing input application rates [Heady et al. (1956), Knetsch
(1959), Miller and Boersma (1966)].
Rarely, water and fertilizer
application regimes are also studied for their effect on yield
[Minhas, Parikh and Srinivasan (1974), Stegman (1980)].
These response functions are highly specific to the experimental conditions and are often criticized for leaving out other
necessary husbandry inputs such as fuel, capital, labor and management.
Since outputs of husbandry process are not directly observable such a
process function
cannot be directly estimated.
But in leaving
them out, the resulting specification error bias often reduces
the applicability of these functions for policy analysis.
There
are some exceptions to this which will be dealt with later.
Reduction of Multicollinearity
On the other hand, mos'
aggregate production functions utilizing
more general data, have their own problems.
has been already discussed.
Aggregation bias
Another problem, somewhat aggrevated
by aggregation is the ;iigh degree of multicollinearity among
the aggregate factors [Ruttan (1965)].
This produces inflated
variances of the estimated coefficients; and the variable
is recognized as leading
Brown (1973)].
to
deletion
specification bias [Hoch (1967),
To overcome this problem, ridge-regression has
been successfully used [Brown (1973), Brown and Beattie (1975),
Hoerl and Kennard (1970)].
Mixed estimators are also popularly
employed [Theil and Goldberger (1961), Holloway (1972), Mittelhammer
37
arid Price (1978), Mitteiharner and Baritelle (1977)].
However,
to the extent possible, multicollinearity should be reduced by
performing aggregation along homogeneous processes i.e. along
separable inputs.
This will reduce the risk of serious multi-
collinearity, though the risk will not be eliminated entirely.
Process Mix Optima and Sadan's Partial Production Functions
The crop growing activity may be broadly separated into the
biological and husbandry processes.
Inputs such as water,
fertilizer, pesticides and herbicides, go into the biological
process, while fuel and services of capital , labor and irri-
gation go into the husbandry process.
The final yield, Y, is
therefore a function of the biological process output and the
husbandry process output.
Symbolically, we have, Y = F(Xl,...,XN)
arid (Xl...XN) is the input set, such that (X1,...,X) can be
= X and (Z1Zm) =
separated and renamed into two groups
.
The independence of the two processes implies that we can write,
Y
F(Xl,...,XN) =
[f(X1,... ,X), g(Z1,.
..
where, f(X1...X) and g(Z1Zm) represent the process outputs
from biological and husbandry processes separately.
It may be noted that specifying,
processes to be either complements
[f,g], allows for the
or substitutes to one another.
Thus separability in this case allows us to treat the yield Y as
a function of two process outputs f arid g.
Consequently graphical
analysis of production efficiency can be applied to this case
with some modification.
-function may be represented with
The
a set of convex isoquants in the f-g space as shown in Figure 1(a).
(b)
(a)
f
uant
iso
9
Figure-i:
In
Figure 1
(a)
Iso-quants
(b)
Iso-costs
(b) above, the isocost curves are arrived
at by considering the apportionment of a given cost dollars into
biological and husbandry process and considering the individual
process-cost functions.
Since separability allows for two-stage
optimization, we can define the process-cost functions under the
following
optimization rule.
For a given process output f,
say, the minimum cost may be derived for a given set of factor
prices.
Thus each process-function has a dual process-cost function.
The four quadrant analysis on the left below indicates the basis
of deriving the isocost curves in f-g space (Figure 2, a).
The
iso-cost curves near the origin are convex (to the origin) because
(a)
39
(b)
f
isocosts
c( f)
i soquants
f
process
T
Cost
Budgets
Figure-2:
(a) Four quadrant analysis of iso-cost lines
(b) Optimal Process mix
each of the individual process-cost functions
concave sections near the origin.
(f) and
(g) have
The important conclusion is
that at lower levels of output, there is a likelihood of multiple
process-mix optima, even when the factor prices are fixed.
is shown on the right above (Figure 2, b).
This
Note that, with the
same technology and factor prices there can be many optimum processmixes.
This indicates the extent of diversity that can be generated
under separability and process substitution.
There is however one exception to be noted here.
If for some
reason, the processes under consideration are perfect complements
40
to one another, then the optimum process-mix is unique for all
levels of final yield.
This is shown below (Figure 3).
ptinium mix
- - Y9
-
\c
g
Figure-3:
Sadan Model :
Perfect Process Complementarity
This situation has already been recognized in the literature
by Sadan (1970).
The process-functions in this situation have been
termed Partial Production Functions by Sadan.
If we replace the
general p-function described above by a Leontief-technology in
f-g-space i.e.
Y =
[f(X), g(Z)J = min[f(), g(2)]
then, we see that we obtain the Sadan-type partial production
functions.
Thus, Sadan type separability puts a further
restriction on the general separability specified by the c-function
The restriction is that the biological and husbandry processes
are perfect complements.
As will be shown later, this type of
separability may be tested for along with the general separability
of the
-function.
41
One important implication of this type of separability under
efficient production is that the specification error bias incurred
in estimating the biological process function without consideration
of husbandry inputs would be zero.
So estimated response functions
are unbiased if efficient production occurs under the Sadan type
technological structure, and they do accurately represent the
biological process functions.
Therefore,
we may conclude that
the biological process functions can be estimated using biological
inputs only, when there exist perfect complementarity between biological and husbandry processes, and production is efficient.
The General and Specific Hypotheses
The hypothesis of week separability appears plausible in
agriculture.
There is almost always a biological process in operation
in conjunction with a non-biological husbandry process.
If we take
wheat as an example, the direct inputs such as water, fertilizer,
pesticide and herbicide enter the biological process of growth.
Growth is also influenced by the environment:
and ambient temperature, etc.
rain, sunshine,
In conjunction with this process, there
is the husbandry process utilizing indirect inputs; these inputs
are generally services from machinery capital, irrigation networks,
labor and management.
These hypotheses can be summarized mathematically in terms
of a separable production function, Y = p[f(X), g(Z)], where X, Z
are biological and husbandry inputs, respectively.
The presumption
42
of this weak separability may be made more explicit by statinq the,
Leontief condition:
(/)j'1 = 0, and the Hicksian type of
X.)
process complementarity:
\> 0.
f
3g1
A more specific thesis may be the assumption that, the biological
and husbandry processes are perfect complements (i.e. no substitutability between the processes).
Under such an extreme assumption,
the partial production functions are equivalent to the response
functions.
This hypothesis may be mathematically stated as,
Sadan complementarity: I_______
[f
=
The above hypotheses appear very plausible, though, they have not
been tested so far in agriculture.
Sadan's own analysis of Partial
Production Functions were carried out under this perfect complementarity assumption, without testing it.
Thus it becomes necessary
to test separability along these 'identifiable" sub-processes
of a crop growing activity.
Sadan (1970) was very likely aware
of this need (See pp. 64).
If the separability hypotheses are empirically ascertained
to be correct, only then
it becomes valid to estimate the response
and the production functions for various farming activities.
These functions for crops, dairy and fodder activities can be
estimated using consistent aggregates obtained by aggregating along
the separate processes.
A value added farm production function
could then be efficiently estimated without aggregation bias.
III.
METHODOLOGICAL BASIS
The basis of performing the separability test lies in the properties of the functional form.
However, when the functional forms
are themselves arbitrary, the test of separability also becomes
arbitrary.
There appear to be two ways to resolve this problem.
One approach would entail going further into the analysis of the
production functions at the micro level.
That is, one can perhaps
determine the growth-functions for plants and animals, based on the
biological laws and begin to build up the macro-level relations from
them.
The other approach would entail approximating a general form
locally, using Taylor expansions.
The advantage here is that locally
a very general form is assumed; the disadvantage is that the results
obtained are mostly local.
Theory of the Test of Separability
Under the optimization hypothesis, there exists duality between
the cost functions and the production functions [Shephard (1970)].
This has allowed inferences on the production parameters from the
cost parameters.
The test on weak separability can also be attempted
using this duality.
If the production function is homothetic and
the cost-function dual to that production is assumed to have a
44
TR.ANSLOG form, the cost-shares are linear functions of the factor
prices (in logarithms)and the output (in logarithms) [Binswanger
(1973)].
Such linear cost-share functions and direct production functions
have been popular in weak separability test [Berndt and Christensen
(1973), Blackorby et al. (1977), Denny and Fuss (1977), and Applebaum
(1977)].
Besides, the homotheticity assumption,
the cost-share or
derived-demand approach requires the use of factor prices.
Generally,
natural resources are characterized as 'public property' and
they are not traded in the market but are allocated on an institutional
basis, they do not have a market price.
Thus these approaches are
less useful in measuring the factor productivities of natural
resources such as water.
The direct production function approach appears more suitable
in productivity measurements of 'public property' type inputs.
In
agriculture, the major 'public property' type of input is irrigation
water, and it often lacks a market price (since no open market exists
for it).
Thus the direct production function approach may dominate
other approaches to measuring factor productivity in agriculture.
Since the Leontief condition for weak separability may also be
simplified as,
i
jk
j
ik = 0,
we see that, weak separability is related to the properties of the
second order cross-partials,
ik'i
constant, for all
i
vis a visthose of
p, that is,
= 1,2,...,n and k = 1,2,... ,m.
Thus
45
a choice of functional form has important implication concerning
separability.
A choice of Cobb-Douglas (CD), Constant Elasticity of
Substitution (CES) or Log-linear (LL) functional form would automatically predispose us to Strong Separability (SS) as a maintained
These functions are unsuitable for testing Weak Sepa-
hypothesis.
rability (WS) assumptions, since SS implies WS.
Flexible Functional Forms and TRANSLOG
Thus for an empirical test for the WS assumption a more general
form than Log-linear or CES is necessary.
This requires the use of
flexible functional form functions, such as, Quadratic (QUAD),
Transcendental Logarithmic (TRANSLOG), Generalized Leontief Linear
(GLL) and higher polymonials (POLY) to test the WS hypothesis.
All
these functions are linear in parameters and OLS procedures can be
employed in their estimation.
Other examples of linear in parameters functions suitable for
testing the WS assumption would be Diewert's (1973) Generalized CobbDouglas (GCD) and McFadden's (1978) Generalized Concave (GCON)
functions.
The many non-linear in parameter forms are less useful in
econometric estimation because of the nonlinear estimation procedure
they require.
Presently, attention will be placed on linear in para-
meter forms only.
Alternatively, nonlinear in parameter production functions can
be employed to test for weak separability.
The nonlinear in parameter
forms are analyzed primarily in terms of the AES.
Fuss, et al.,
46
(1978) discuss various such functional forms.
They have also gener-
alized variants of the Cobb-Douglas and CES functions [Fuss et al.,
(1978), p. 242].
The most frequent functional form used to test the WS assumption has been the TRANSLOG [Berndt and Christensen (1973 and 1974),
Denny and Fuss (1977), Corbo and Meller (1979)].
The main advantage
of the TRANSLOG form is that it can be used to approximate any general
function at a point up to a second-order Taylor expansion.
Thus the
test of the WS assumption using a TRANSLOG approximation (TRANSLOGAP)
is a test of WS at a point in the factor space (X,Z).
For a global
test of WS, exact specification of a production function becomes
necessary.
Unfortunately, such exact TRANSLOG specification
(TRANSLOGEX) is not without its own problem.
Exact Translog and Translog Approximation
Denny and Fuss (1977) and Blackorby, Primont and Russell (1977)
have shown that the TRANSLOGEX does not have enough flexibility to
specify weak separability and non-linearity together.
proved that (1)
They have
is Log-linear in f(X) and g(Z) when f(X) and g(Z)
are Log-linear in X and Z, respectively.
Thus as Denny and Fuss
(1977) have argued, the global test of Berndt and Christensen (1973)
using TRANSLOGEX is not merely a test of the WS assumption but
also a test of inflexibility of functional form as indicated
47
above.
The TRANSLOGEX specification generates inflexibility, that
either
is LL or f and g are LL.
(Proof in the Appendix-A).
The TRANSLOG production function is given as,
NN
N
+ E
in'! =
where, y..
The
use
lnX1 + 1/2
c
lnX
iflXj
1,...,N.
i,j
of the WS assumption, provides the following Leontief
conditions for the TRANSLOG.
N
cJ1ik) +
E
m=i
where, m = 1,... ,N.
(. y.
im jk
jm ik
)
lnX
m
= 0
This leads to a sufficient condition, called
the Linear Restriction of Berndt and Christensen, as,
for
e X and Zk e Z.
leads to a
g(Z).
1jk
1ik
= 0
This linear separability restriction
-function which is Cobb-Douglas in TRANSLOG f(X) and
(See Appendix A).
Alternatively, the necessary and sufficient condition for
WS to hold is that,
ijk
j1ik
for m = l,2,...,N.
0, and
im"jk
'jm
.
ik
= 0
These imply that,
= ''ik -
for m;k
jk
= i,2,...,N.
jm
This nonlinear-restriction, according to Denny and Fuss (1977)
and Blackorby et al. (1977), implies that the q- function is TRANSLOG
in Log-linear
f(X) and g(Z).
The nonlinear separability restriction
implies that the elasticity of substitution within the process
inputs X.
is always one.
and X
The linearity restriction on the
other hand allows for non-unitary, variable elasticity of substitution
between the process inputs X.
and X
etc. at the cost of constraining
the elasticity of process substitution between f(X) and g(Z) at
unity.
Therefore, rejection of WS with TRANSLOGEX could result
from the assumption that (a)
X1
i.e.
X
= 10, or, (b)
l0.
f and g, i.e. afg
the elasticity of substitution between
the elasticity of substitution between
In order to avoid this problem, a TRANSLOGAP
specification is needed, though this does not allow for a global
test of WS.
Other Forms and Monotonic Transformation of Variables
The separability property holds for all monotonic
transformations
if F(Xl,...,XN) is made weakly
of variables, i.e.,
separable, then
y
separable when x.
is a transformation of X., such that, .! > 0
1
G(xl,...,xN) can also be shown to be weakly
xi
or
< 0.
(Proof in the Appendix A, Theorem_i).
One very
Xi
important implication of this result is that, the results derived
for TRANSLOGEX
forms.
above
holds analogously for GLL and Quadratic
For the Quadratic, linear WS restrictions imply that the
elasticity of process-substitution,
fg
is infinity (See Appendix
A, Theorem-33) because G(xi,...,xn) is a linear function of two
(or more) process-functions f(x) and g(z), which are quadratic
49
in x's and z's.
The elasticity of substitution between x1 and
Xj within the process f however is variable but finite.
With non-linear WS restrictions, the Quadratic G(x1,..
.
becomes a quadratic function of linear f and
g.
,x)
The elasticity
of Process-substitution.Gfg is variable and finite but the elas-
ticities within the processa
become infinitely large.
The sanie
results can be extended to the GLL specification [Blackorby et al
(1977)].
This indicates that TRANSLOGEX is a somewhat more general
specification of any exact quadratic, in the following sense
it allows finite elasticity of substitution between processes under
the nonlinear WS restriction and unitary elasticity of substitution under linear WS restriction.
Econometric Estimation and Test of Hypotheses
Corbo and Meller (1979) have used the direct production function
approach in estimating the manufacturing sector production functions.
They also used the TRANSLOG flexible form to test for weak and
strong separability.
Their results indicate that in the majority
of cases the Log-linear
(strong separability) assumption could
not be rejected; therefore, their results indirectly support
the
aggregate industry production function using aggregate inputs.
In the present study,
The TRANSLOGAP specification is used
because, (i) it is linear in parameters and it provides a second
order Taylor approximation to a general function (at a point)
50
and (ii) it is flexible enough to permit a local test of weak and
strong separability.
The global weak separability test could be
performed using (i) higher order Taylor approximations and (ii)
Nonlinear in parameters forms.
Both methods above have their own drawbacks.
Cubic and forth
order approximations become cumbersome because of the rapid proliferation
of
parameters.
The nonlinear in parameters forms
on the other hand are computationally cumbersome since they need the
use
of
iterative estimation procedures.
Futhermore, these
nonlinear estimators lack the BLUE properties of OLS estimators
of linear in parameter models.
Even local tests of weak separability require
performing
the tests on the validity of the linear and nonlinear restrictions
imposed on the TRANSLOGAP parameter (See pp.
restrictions
i.e.
''ik
47).
The linear
are equivalent to block-diagonalization of y-matrix
e X, Xk e Xr where XS[. Xr
0,
the input set.
0 and xstJXr = X,
TRANSLOGAP can be written as,
-
yc0+ctX+aX+ x'YX+X'YX
where, ct = (ct,,.. ,c), x
(x1,... ,XN), and y
ll"lN
I
,
jN5
or alternatively as,
A: \fr
fx\
y =
+
(
:
c
)'') + (x'
fy.0\(x
xt )k:
)t
\0y1"x
etc.
51
under the linear weak separability restriction (LWSR).
Thus we
see that the linear separability test requires testing the null
hypothesis
H:
y F}
We can estimate the TRANSLOGAP as a linear model Y = X
+ e,
where X represents a TXK observation matrix on K independent variables
and there are K
s-coefficients corresponding to these.
It should
be noted that, K = (1 + N) + 1/2 N(N + 1), where N is the number of
inputs in the TRANSLOGAP production function.
This is so because
there are N linear terms in TRANSLOG form along with N(N + 1)/2
cross-product terms (1/2 due to the assumed symmetry) and one
constant term.
is the usual error term with mean zero and
finite variance, cr2.
Test Statistics for Linear Restriction
The linear restrictions can therefore be tested by applying
the following linear restriction on the linear model representing
the TRANSLOG,
R=Oor °l1°lN
! ?9
9
N+l
0q1
o
qN
1
.... 0J
k
j
52
where, R is a q x K restriction matrix with first N columns of
zeros and the rest K - N columns of zeros and zeros and l's.
the diagonalization requires putting q
If
s to zero, the remaining
K - N columns can be written as a q x q unitary matrix followed
by columns, with zeros,
of zeros with q x
1
The right hand side consists of a column
dimension.
This is a standard linearly restricted model.
under the null hypothesis, H0: R
The OLS estimator
= 0, is given by the following
expression:
b* = b + (X'X)
where
r =
R'[R(X'X)R']1 [r - Rb],
(q x 1) column of zeros in this case; and
X'Y is the unrestricted OLS estimator of the true
b = (X'X)'
TRANSLOG parameters,
.
Therefore, the test of null hypothesis H: R
= 0, can be
performed by using the following F-statistics:
=(2)[(b
F(q
,
where
- b*)(X'X) (b - b*)]
1-K)
S2 is the unconstrained linear least squares estimator
of the error variance, a2 [Theil (1971), p. 143].
statistic
can be simplified to:
F(q,T_K) =(T_K)[(e*'e*
eej -
where, e'e
This test
2,
ii
j
the sum of the squared errors of the non-restricted
T- K
model, and e'e
is the sum of the squared errors of the model
53
restricted under H0: R
= 0.
The statistic
can be shown to be
equivalent to the maximum likelihood ratio test statistic
[Theil (1971)
p. 143 ].,
The nonlinear weak separability restriction
non-linear restrictionon the c
and
(NWSR) also imply
coefficients or the
of the linear model representing TRANSLOGAP.
s
However, these
nonlinear restrictions cannot be incorporated in the linear form
R
The implication is that
= 0.
the previously derived test
statistics cannot be directly applied.
Fortunately however, the
nonlinear restriction can be built into the linear model prior
to estimation; and therefore the nonlinearly restricted model
can be expressed as a simpler non-linear model.
These explicit
non-linear in parameter models can be directly estimated using
nonlinear techniques.
Test Statistics for Non-Linear Restriction
The nonlinear weak separability restriction implies that the
terms
of
and
the
TRANSLOG form can be generated from the two portions
of the vector, a = (al,...,aN)
l'"'N S
)].
r
In Appendix-A, Theorem-2, it has been shown that, if the inputs
are 'nonlinearly' separated into
and
the TRANSLOG production
can be generated as
function
,
cross-product matrix, as follows
I
Ii,;ii\
jc a:i a
)
I
fl,I,
:
ici
act
the original 'y-matrix of
54
This implies that the translog can be simplified as:
+ 111
kk'(')2
') + ax
+
(1)2
+ (kl + 1k) (') (a';)
We note that kk', 11' and k'l + 1k are scalars:
Therefore the
above expression is a nonlinear in parameters equation which
implicitly contains the nonlinear restriction due to NWSR.
The
scalars kk', 11, kl' + 1k' have to be estimated along with
and a using
iterative nonlinear estimators (e.g. Gauss Method).
Let the nonlinear least squares model be represented by
y = f(X16) + , where, e'N(0,a2),
(explicit) function of
and X.
where, f(X1) is the nonlinear
Now let the nonlinear least
be s*.
square estimates of the true coefficient
Let
2
represent
the nonlinear least squares estimate of error variance, a2.
Under
*
some 'regularity' conditions, both t3* and a
maximum likelihood estimators.
are consistent
[Judge et al. (1980), pp. 727].
Further, these estimates are assymptotically normal with mean
and a2, and they are assymptotically efficient.
Thus
under the
estimators
ML
and
estimators g and
ML
normality assumption, the maximum likelihood
are equivalent to the nonlinear least squares
2
respectively. The variance covariance matrix
is however known as,
Var(ML)
=
aML[Z(ML)
Z(8ML)]
55
where, Z(BML), is defined at true parameter value
as,
,
[See Judge et al. (1980) pp. 725-727.]
= Z(ML).
As long as f(X,) meets some regularity conditions, i.e.,
differentiability and invertability of [Z()'Z()] matrix,
will be consistent and also asymptotically normal.
ML
This allows
us to define,
ee
**
= SML(ML) =
f(X,ML))'
(
f(X,ML))
as a sum of independent and identically distributed asymptotic
This means that ee*/cy2 is asymptotically
normal error terms.
X2-distributed with T-K+q degrees of freedom (where q is the number
of nonlinear restrictions implicitly employed on K
s in the
original linear model).
This allows us to show that
2
SML(3ML)
=
T-K+q
T-K+q
is a reasonable estimator of true
Now, (e
e
-
2
[Judge et al. (1980), pp. 725.]
e'e)/a2, the difference of two y.2-distributed
variables, is also a
2-distributed variable,
true only asymptotically.
though this is
The degrees of freedom of the combined
variable is (T-K+q) - (T-K) = q.
Thus we may again take the
ratio,
T-K
F=
q
(ee4
\
-
e'e
ee".
I
We note that the denominator is
56
x2-distributed with T-K degrees of freedom, and the numerator is
also asymptotically x2-distributed with q degrees of freedom.
Therefore, the ratio, F, above, is asympotically F-distributed
with q and T-K degrees of freedom [Theil (1971), pp. 80-81].
Therefore, in conclusion, we see that nonlinear restrictions
cannot be tested using the regular F-statistics derived under
linear restriction.
In contrast to the linear restriction model,
where normality assumption about the disturbance term was unnecessary,
the non-linear restriction model requires normality of the disturbance
terms before a test statistic can be constructed.
In view of this,
nonlinear restrictions are not as easily tested as the linear
ones.
The results of the test also need to be interpreted as
asymptotically valid, and only under the normality assumption.
Negative Random Error Model Under Sadan Complementarity
When perfect complementarity between husbandry and the biological process holds, even under a wide range of relative factor
prices, the process-mix will remain stable.
This is due to the
Leontief (input-output type) linear technology isoquants in the
f-g-space as shown below:
i soquants
opti
mi
I socosts
Figure-4:
Sadan Model
57
This also implies that the regular econometric approach fails,
because y =
(f,g) = Mm
in f-g space.
(f,g) is no longer a continuous function
The function can be indicated in three dimension
as the lower parts of two intersecting planes as shown in Figure 5.
0
V
mal mix
yc
f
0
f0
Figure-5:
Perfect Process Complementarity
and Discontinuous Yield Function
Plane OAD represents the potential of the husbandry process.
Fur a given level of the process output, say at g0, the maximum
yield is y0.
The diagram clearly indicatesy0 to be indepen-
dent of the biological process output f, along MN, as long as
f is greater than f.
When f falls below f0, the yield does
not follow MN line, but follows the line common to MNPQ and OBC
planes.
Thus, the feasible points are on or under the ridge
represented by two planes OBX and OAX.
The points on the planes OXC and OXD are not feasible, except
on the line OX.
Under the assumption of efficient production,
we expect the producer to regulate f and g such that the yield
is always on OX or near it.
Thus as far as, one single process
output f and its relationship to yield y is considered, the possibility of y being equal to f depends upon whether g-output is
constraining or not.
This may be summarized by saying that
y
where,
=
f(X) +
is always negative or zero, e
<
0.
Let us say that
is a random variable with a negative expectation, E(e) =
and variance V()
=
<
.
<
0,
Under these assumptions, we will
find that the true model coefficients except the constant term may be
estimated by the least squares method.
The coefficient estimates of the linear in-parameter form
for f(X) will be seen to be unbiased except for the constant term,
The estimate of
will however be biased to a value,
v=
Let us define a new error term
has a zero expectation, E(v) = E(e
6
+
, this random variable
-
E(e)
and the variance remains unchanged, V()
= V(e)
-
i-i
V(p)
=
= 0.
V(e).
Therefore the negative error model may be transformed into
a
general linear model,
y = XB + e =
Xc3
+
+
Note that the constant term
in 3'
=
(
01 ... .
changed to represent a new coefficient vector, b'
The difference between
of the first term,
+
,
so that
'
k
)
can now be
+
and b' is only in terms
y = Xb +
= 0, v()
,
holds and we can use OLS to estimate, b, using the matrix equations,
= (x'x)
(XV).
We note that, all the coefficients
I''3k are
estimated without
bias by bl,...,bk respectively, because
E(b) = E[(X'X)
(xExb + X)] = E(b) + 0
o-
We also note that E(b0)
and therefore the constant term of
the regression will be downward biased, by the unknown quantity
.
The same arguments could also be repeated in relation to the
g-function under Sadan complementarity.
Thus under perfectly
complimentary processes, the process functions can be estimated
except for the constant terms.
The estimates of the marginal
productivities of the inputs used will however not be biased.
The factor productivity coefficients will be asymptotically normal.
The problem however remains as how best to test for perfect process
complementari ty.
The negative error models have already been used in the
description of production frontiers [Judge et al. (1980) pp. 302,
Aign2r et al. (1976), Lee and Tyler (1978)].
Alternative speci-
fications for the negative error terms are possible, but does
not seem to provide any extra benefits beyond what has been already
gained.
Schmidt (1976) has attempted maximum likelihood (ML) estimator
by assuming half-normal distributions for e,
f(e)
2
V2io
exp
f
2
\
,
e < o.
2)
He indicates that one of the regularity conditions required for
maximum likelihood estimation is violated under half-normal specification.
Under this situation, the ML estimates are possible
but their sampling properties are uncertain.
Another possibility
is to use two error terms, though this would imply two random
processes at work, which is rather doubtful [Judge et al. (1980)
pp. 302].
Sadan Model and a Simple Cubic Approximation
The previous discussion on the Sadan function, y = miri(f,g),
indicated that the function is represented as a slanted roof ridge
over the f-g plane, shown below.
Though this ridge is discontinuous
(a)
(b)
f
i'A
optimal process mix
x
;oquants
f
Figure-6:
Discontinuous yield-surface in Y-f-g space
61
along OX, a number of surfaces can be approximately fitted over
this roof-ridge, particularly when most of the observation points
are scattered along OX.
One possibility is to approximate this
ridge with the inverted inclined cylinder with a parabolic crosssection as shown below:
y
process
Figure 7:
Inverted Inclined Cylinder
From the theory of conic sections it can be shown that such a
surface can be represented in the positive f-g quadrant with the
following quadratic in f and g:
y = (f + g) where,
ii
(f - g)2
is a positive constant.
It is also known that increasing
the power of the second right-hand term in the above equation
sharpens the top part of the inverted parabolic cylinder(Figure 8).
(a)
(b)
I
1
g
;imal process
mix
i soquants
0
1)
Figure-8:
(a)
Isoquants
f
(b)
Inverted Inclined Cylinder
62
On the other hand decreasing the power of the second term flattens
the ridge of parabolic cylinder, see diagram below:
(a)
(b)
//
7/f
I
f
Figure-9:
(a)
/
-
optimum
mix
0
Isoquants
(b)
Inverted Cylinder
Now, if f and g are quadratic functions of X and Z respectively,
Y = (f + g) - rj(f - g)j312 represents a cubic expression.
Thus
if Sadan complementarity exists, the sharp ridge of the function,
y = mm
The
(f,g) may be approximated with, y = (f + g)
nl(
g)1312.
test of Sadan complementarity may be performed estimating
the equation above and acertaining
truly
if
n and other coefficients
represent an inverted parabolic cylinder surface.
when the function,
y =(f + g) - n(f - g)!312
is estimated using a general quadratic for f(X) and g(Z), the
63
coefficient n should be positive and statistically significant.
The test would of course require a nonlinear function estimation
and the use of asymptotic properties of these nonlinear estimators.
In contrast to the negative random error models, which can
estimate either process function this simple cubic approximation
allows the use of errors due to f and g simultaneously.
From this
standpoint, the simple cubic is better suited to test perfect
complementarity. The negative random error model,on the other
hand is not capable of providing a test of perfect complementarity.
Nested Hypotheses Sequence
The hypothesis to be testedcan be group-wise linear and nonlinear weak separability in the context of a translog function.
More restrictively, we can have complete pair-wise strong separability
and additive strong separability.
Both of the presently discussed
separabilities imply weak separability but not vice versa.
Simi-
larly,we may have further subdivision within each category, using
linear homogeneity.
This allows us to create and test a sequence
of nested hypotheses.
For instance, if the strong separability hypothesis is not
rejected statistically, it becomes unnecessary to test for weak separability.
This is true when linear homogeneity and strong separability
is not rejected.
But even if strong separability is rejected
there may still exist weak separability, so this needs to be
64
Linear homogeneity may be tested for at the first stage.
tested.
(See Figure 10).
For the translog function, linear homogeneity is equivalent to
the following linear restrictions on the ct's and
n
10
.
1
i=l
=
E
00 for all
j=
l,2,...,n.
One may test for linear homogeneity using the linear restriction
matrix, R
=
r, as discussed above and applying the F-statistics to
test the null hypothesis H0: R
=
r, representing the equations above
[McFadden et al. (1978)].
Since monotonicity and quasi-concavity are not always ascertained
for the translog, they need to be confirmed at some stage.
If strong
additive separability and linear homogeneity becomes the nonrejectable hypothesis, then the Cobb-Douglas form is implied and monotonicity
and quasi-concavity is assured.
Except in this case, the translog
needs to be tested for both monotonicity and quasi-concavity over the
relevant range of factor inputs.
The test of monotonicity at a given
factor combination point may be represented by the following null
hypothesis,
H0:
(ct
x1)
il
0.
65
TRANSLOG APPROXIMATION
+
(1) TEST LINEAR HOMOGENEITY
H0: Ec
LINEAR HOMOGENEITY
0
1,
3
(2) TEST COMPLETE
PAIR-WISE SEPARABILITY
H:
= 0, if i
Complete Pairwise
Complete Pairwise
Separability
Separability
j.
Yes
No
Cobb-Douglas
Yes
Cobb-Douglas
N
(3) TEST GROUP-WISE
Linear Weak
LINEAR WS RESTRICTION
H0:
1ik
Separability
XS, Zk e Zr.
= 0, if X1
L
(4) TEST GROUP-WISE NON-LINEAR
Nonlinear Weak
WS RESTRICTION
Separability
for all
1
=
r for all
1
= l,...,N.
= ''il
0
u.
3
Y .
jl
and q 1
=
=
q
1q1
TEST MO NOTONICITY
H
:
0
(
.
1
+ E1
1i1
Monotonicity
I
Monotoni city
Monotoni city
>
) TEST QUASICONVEXITY
H0: X'HX < 0.
Quasi-convexity
41
H is the bordered Hessian
of
END
END
(f,g) at the point x.
Figure-lU:
Nested Hypothesis Testing
END
The test of quasi-concavity requires the bordered Hessian of the
translog production function to be negative definite at the given
point.
This requires that, x'H x > 0, where H = the bordered Hessian
of the translog form [Corbo and Meller (1979)].
In Figure 10 above the sequence of tests on the nested hypothesis
If c.% level of sequence of significance is allotted
is indicated.
to each level of test, there being 4 levels of these tests, we have
So if c = 0025, 4a = 0.10, and the
the overall significance of 4ct%.
overall significance is still 10%.
The test would be a sequence of
F-tests derived for the restriction for each level.
Single and Multiple Partitions Separability
Single and multiple partition separability is a generalization
of the two extremes, complete pairwise separability and simple weak
separability, into two groups.
For example, let us consider a
function of three variables, x1, x2 and x3.
following partitions on these:
We can stipulate the
three different single-partitions
[(x1, x2), x3], [x1, (x2, x3)], [(x1, x3), x2], and one doublepartition [(x1), (x2), (x3)].
in fact, inseparable.
Let us further assume that x1 and x2 are,
If we test for all three single partitions and
one double-partition, statistically, we expect three rejections and one
non-rejection.
The rejected restrictions would be [x1 (x2, x3)],
[(x1, x3), x2] and [(x1), (x2), (x3)].
would be [(x1, x2), x3].
The non-rejected restriction
67
This suggests
that if the objective is merely to confirm
separability of x3 from (x1, x2), we may simply test this particular restriction.
On the other hand if the objective is to test
for any existing inseparabilities, we may have to perform all of the
tests.
The first test would appropriately be the most restrictive
i.e. the double-partition (or global or pairwise separability),
[(x1), (x2), (x3)].
If this is not-rejected in a statistical
test, further separability tests are unnecessary.
At the other
extreme, all the restrictions can be rejected, indicating no
separability, and the non-existence of Composite inputs or processfunctions.
In the case of three variables, if two single-partitions
are not rejected, for the sake of logical consistency, we require
that the double-partition be non-rejected also.
spurious results can be expected
But practically,
where the choice between the
non-rejects need to be made.
Extending these arguments to four-variables, we first
test for complete pair-wise separability.
If this test is rejected,
then we may test for double-partitions; six of them:
[(x1, x2), x3,
x4J, [(x1, x3), x2, x4], [(x1, x4), x2, x3], [(x2, x3), x1, x4J,
[(x2, x4), x1, x3] and [(x3, x4), x1, x2J.
If all of these six
are also rejected, then we have total inseparability amongst
[(x1, x2, x3, x4)].
If only one is not rejected out of six possi-
bilities, this would be strong support for considering that there
exists an non-separable pair amongst four x's.
However, if more
than one partition become not rejected then logical consistency
would require that there is
global pairwise separability.
When
direct test of global pairwise separability is rejected, but two
or more of double-partitions are not rejected, a contradition
arises which may not be resolved with the given data.
In such a situation (except when one (or more) of the
non-rejections are marginal) further choice between the nan-rejects
(and therefore mutually competing notions of separability) is
difficult on purely statistical grounds.
Thus in conclusion, we must note that
though a complete
rejection of separability is always possible, the reverse, where
a specific type of separability alone is non-rejected, is possible
but not always guaranteed.
The data set may or may not be able
to distinguish between two competing notions of separability.
When such a case arises, the choice can be made only on the basis
of some extraneous information outside the model and data.
One
possibility is to extend the order of approximations of a function,
that is, tomake a cubic out of the quadratic, and attempt to
resolve the choice to a single model of separability.
In models
involving many variables, such a procedure, however, rapidly
increases the number of parameters to be estimated.
Another
possibility is to use nonlinear in parameter functions with appro-
priate restrictions on AES to represent separability (along with
Sadan type restrictions).
4!]
IV.
MODEL SPECIFICATION AND HYPOTHESES
Model and Maintained Hypotheses
The validity of the biological and husbandry processes will be
ascertained using separability tests on wheat production function.
The model will be a general function locally approximated by the
TRANSLOG function.
The model is as follows:
Y = G(Q, B, F, P, T, I, N, K, M).
where, Y is per acre wheat yield, B is pesticide per acre, F is fertilizer per acre,
I is percent of irrigated acres, P is precipitation in
inches, T is temperature index, N is labor measured in dollar wage,
K is capital service measured by fuel and oil expenditures.
M repre-
sents the management input index and Q represents the land quality
index.
It may be noted here that there are no data available for
Q, B, and M.
They are listed here merely to indicate that they should
ideally be included in a production function for wheat.
The model could also be specified in terms of total yield
Y =
(Q, B, F, P, T, I, N, K, M, A)
where the variables B, F, N, K are corresponding total inputs, Q, P,
T, M are corresponding indices,
wheat acreage.
I
is irrigated acres and A is total
70
Since cropping activity is spatially replicable, we know
the
(..) function will be linear homogenous in inputs F, I, N, K
The function G(..) corresponds to the model described by
in the sense that, G(..)
be obtained from
(.j,
(..), by dividing
the total input and output variables by the total acreage A.
The assumption that
(..) is linearly homogeneous will therefore
be a maintained hypothesis in the present study.
This 'constant returns to scale' assumption is reasonable
for a crop production function.
The same assumption would not
be valid if other activities were also present such as forage
and cattle (or dairy).
If there exist externalities between
processes, the value added function may obtain increasing returns to
scale.
There is some evidence of increasing acreage size of the farms,
indicating an existence of economies of scale.
This acreage growth of the farms may result from increasing
returns to management inputs in 'management processes' and/or
because of increased efficiency of capital use.
Bigger and better
machines may result in substantial reduction in labor and management
inputs (for the same output) and allow for scale increases by
relieving the slack created in management capability.
When 'pro-
cesses' are exact replicates, management inputs can show increasing
returns to scale, because almost the same operations on the management
side (e.g. procurement of inputs, keeping accounts and managing
the inventory) result in large direct input
utilization.
71
The issue of the growth of farms in recent years, though important,
is not the focus here.
The issue appears to require a more detailed
study of the management process, which at present is lacking.
The
problem here arises from the fact that management inputs are
diverse in quality, and there is no readily available index or
quantified variable to represent it.
This particular problem
does not appear to have been satisfactorily dealt with so far.
Another aspect of the problem consists of the fact that 'entrepreneural ability' is rather immobile between farms and is not
marketed readily.
Proposed Hypotheses
It is hypothesized that any crop growing activity may be
conceptually looked upon as consisting of the biological growth
process and the husbandry process.
The husbandry process creates
a suitable culture medium by tillage, seedbed preparation, irrigation etc. so that the biological process may succeed.
maturity of plants,
the husbandry process continues again in terms
of harvesting, drying etc.
cesses imply that
And after
The sequential nature of these pro-
the biological and the husbandry processes
are economically separable.
Thus it is hypothesized that,
Y = G(Q, F, P. B, T, I, M, K, N)
can be written in the following separable form
Y = cp[f(Q, F, P. B, T), g(K, N, I, M)J
72
where, f is the biological process function and g is the husbandry
function.
If, the irrigation input were measured in terms of acre
feet of water instead of percent of irrigated land, it would be an
argument in the biological process function. However, when irrigation
represents a service from
irrigation capital (or infrastructure
of pump sets and distribution pipe lines or the culverts and ditches),
its logical place appears to be in the husbandry function.
Operationally, therefore, it is hypothesized that the per
acre model will be weakly separable into at least two groups
consisting of the weather/biological variables Q, F, P, B, I and
the husbandry inputs K, N, I and fri.
It is possible that all of the
inputs are strongly separable from one another; this would still be
consistent with the notion of two process function f and g in
-function.
If, on the other hand, a pair of variables, one be-
longing to one group (say, biological process) and the other
belonging to the other (say, the husbandry process) are found
to be empirically inseparable, then this would mean an empirical
refutation of the notion of those two subprocesses.
Even that
however does not exclude the possibility of having new processes
redefined within the production scheme.
There is a possibility
of regrouping inputs into new separable groups, each group corresponding to a 'news process.
Such empirically constructed 'processes'
may or may not have their theoretical justifications.
Therefore, one
should start with the a priori theoretical possibilities of
73
subprocesses and weak separability, rather than reasoning from
'empirical separability' to 'processes'.
For example, in the case of cropping the knowledge of tillage,
seedbed preparation, plant growth, harvesting as sequential acti-
vities leadone to assert input independence amongst these activities.
So we hypothesize weak separability between fertilizer, water,
pesticides and fuel, labor, capital,etc.
When such intimate
knowledge of the detail of the subprocesses is
not available,
it becomes necessary to allow the empirical identification of
separability to lead the way to development of the theory.
Such may be the case of management inputs, if these inputs
have satisfactory indices for them.
Though management process
outut is not tangible, it is possible that certain attributes
of the manager and the organization are good indicators of their
effectiveness.
Very often these attributes are vague,
though each
of them stand a better chance of being quantified rather than
"management" itself.
Their effectiveness is seen (often) clearly
when they appear simultaneously.
In the absence of such indices, the major alternative appears
to be as follows.
If the management inputs (attributes) are
assumed separable from other inputs, and there exists perfect
complementarity between 'management process' and the rest of
the 'processes', then, the partial production function of the
remaining 'processes' may be estimated without specification
bias, provided, efficient production is assumed.
74
Symbolically, if h(M) represents the management process output,
then the yield is definable as,
Y
[f(X), g(Z), h(M)]
under the weak separability assumption.
Again, f(x) represents
biological process function and g(Z) represents the husbandry
process function.
The perfect complimentarity between h, and,
f and g, can be represented as,
Y
min[p(f(X), g(Z)), h(M)].
Under the assumption of efficient production, we have,
Y
(f(X), g(Z))
h(M).
Thus, only if we assume that
weak separability with perfect
process complimentarity between 'management' and other 'processes'
exist, along with efficiency in production, it becomes valid to
estimate the crop-function.
Otherwise specifiction bias results.
This is another maintained hypothesis in this model.
The husbandry process inputs, such as capital service, K,
and labor, N, are assumed independent of the weather variables,
precipitation, P. and the temperature index T.
Furthermore,
the biological input, fertilizer, F, is also initially assumed
to be separable from the husbandry inputs, K and N.
On the other
hand, irrigation input may be dependent on P and 1, because P
can substitute for irrigation water.
be empirically tested.
The later possibility has to
75
Separability between F and K and N can, however, be tested
afterwards if P and T are indeed separable from F, I, N, K.
The
separability structure then simplifies to the three distinct possible
groups, weather, biological and husbandry inputs.
On the other
hand, weather inputs can also be regarded separable from economic
inputs on the grounds that weather variables do not influence
economic inputs because they are not known be-forehand.
It is further hypothesized that in the case of a crop production
process, the biological and husbandry processes are not only
distinct and separable, but they are, in line with Sadans thinking,
perfect complements.
The implication is that the
-function is
reduced to the case where,
Y = Min[f(X), g(Z)].
In other words, the substitutability between f(x), the biological
and g(Z), the husbandry functions. is zero or there is perfect
complementarity between them.
to
This hypothesis is equivalent
weak separability and additionally, perfect complementarity
between f and g, as has already been discussed.
The isoquants
in f-g-space are of Leontief-type,i.e., they are L-shaped.
discontinuity
This
represents a problem in testing for this particular
'nix of hypotheses' in empirical terms, and approximations using
cubic functions may be employed for testing purpose.
76
Regional Production Functions
Eastern Oregon is comparatively dry relative to western Oregon.
This difference in precipitation influences the type of wheat grown
in these two regions.
The low annual precipitation in most of eastern
Oregon is often unable to sustain annual wheat crops.
Thus in eastern
Oregon the common practice consists of winter wheat cultivated every
alternate year.
The moisture accumulated in the ground during a
fallow year sustains the moisture requirement of the cropping year.
In western Oregon, on the other hand, spring wheat is the
dominant class produced.
The difference in variety of wheat grown and
the incidence of precipitation makes it probable that there exists two
distinct production functions--one for eastern Oregon and one for
western Oregon.
The high precipitation in western Oregon implies that
there is little need to irrigate spring wheat.
Irrigation in western
Oregon is expected to affect yield to a lesser extent than in eastern
Oregon.
Alternate year cropping in eastern Oregon also implies that the
input of machinery services per unit of land cultivated there will be
lower than that in western Oregon.
The graph below indicates clearly
that, as far as tractors, the major tillage machinery capital,is
concerned, eastern Oregon employs fewer per acre than does western
Oregon.
This is the primary reason for believing that the number of
tractors does not represent machinery service input into the production
77
GRAPH 1: Cropland Harvested and Tractors
LN (Acres) vs. LN (Tractors #)
1974 Oregon Counties
UI (Acres)
(
6?
Western Oregon
Eastern Oregon
a
a
a
a
a
3.
I
2
r
1
.
I
2
LN (Tractors)
3
4
process.
The 'stock' capital must be replaced with
services concept.
a
fund' of
Capital Service is thus better measured in terms
of rate of utilization.
Summary of Hypotheses
There are these common maintained hypotheses for two types
of models to be summarised later.
(i) 'management process' is perfectly complementary to other
'processes' (such as biological and husbandry)
(ii) production is efficient with respect to the 'management'
and 'other' processes;
(iii) The original 'total' model,F(..),is linearly homogeneous
in total inputs, land, fertilizer,..etc.
The implication of (i) and (ii) is that the 'Other'
process-
function (or partial production function) can be estimated without
specification error.
The third assumption allows one to move
from the 'total' model,
to the 'per acre' model, G(..).
There is a possibility that a) P and T, the weather
variables in the biological inputs are separable from both the
biological input F and the husbandry inputs K, I, N;
Y = p[P, 1, G(F, K, I, N)].
or b) the husbandry inputs N, K are independent of all biological
process inputs,
Y =
Ew(P, T, F, I), v(K, N, I)].
79
Though a) and b) can both be tested, they are 'maintained'
hypothesis in this study.
Thus there are two models, model-A
under (a) and model-B under (b).
The hypothesis to be tested using model-A are (i) Separability
between husbandry and biological inputs,
Y =
[P, T, f(F), g(K, N, I)].
(ii) Perfect Complementarity between husbandry and biological
processes,
Y = min[f(P, T, (F)), g(P, T, (K, N, I))].
The hypothesis to be tested using model-B is that (P, T)
is separable from (F, I), i.e.
Y =
[W((P, T), (F, I)), V(K, N, I)].
Sources of Data
Oregon county level data on inputs and outputs of wheat
were obtained from the Census of Agriculture for the years
1959, 1964, 1969, and 1974.
954,
The data available are total wheat
yield, acreage of wheat harvested, acreage of irrigated wheat, and
the total amount of dry and liquid fertilizer applied.
Unfortunately,
the amount of water applied to wheat is not known by county.
The
husbandry inputs such as fuel (energy), labor and capital machinery
services are not available specifically for wheat.
The average per
acre values in all uses may, however, be calculated from the data
available.
This measure is not entirely satisfactory since fuel
and labor ratios may vary between crops to a considerable extent.
E;I
The source of climatological data for Oregon counties are the
reports from the National Oceanic and Atmospheric Administration,
U.
S. Department of Commerce. [Annual Summary of Climatological Data
for Oregon, Environmental Data Service, NOAA, USDC, 1954, 1959,
1964, 1969, 1974.]
This source was particularly useful since it pro-
vides a yearly summary of precipitation and temperature.
County
level data were developed by choosing the 'representative' weather
station in the county.
Since annual averages are taken, the errors
dur to geographic heterogeneity of precipitation and temperature will
probably be small.
Some error will, however, always remain.
Appen-
dix-C contains the list of weather stations chosen to represent the
counties..
Data Description
Fuel and labor inputs vary from crop to crop.
When county averages
are used, error may be reduced somewhat provided that the averaging
process was conducted with a single crop.
siderably from farm to farm.
to county.
The crops grown vary con-
The crops grown vary less so from county
If the crop-mix between the counties are similar, fuel
per acre (on the average) may be a reasonable measure of the total
service of machinery capital
81
Although capital stock could be employed as a variable, this
too is heterogeneous in terms of horse-power, type of machine, age,
and technical efficiency, and so this measurement of capital input
becomes suspect.
More importantly, 'stock' measures ignore utiliza-
tion rates of these machines.
Fuel on the other hand can measure
the utilization rate accurately.
However, when fuel and oil is
aggregated over different mechanical equipment and operations, this
accuracy might be lost.
Similar considerations on heterogeneity of labor service apply.
However, it is remarkable to note that the ratio of deflated labor
and duel (in logorithms) expenditures have remained relatively
constant for 1954, 1959, 1964, 1969 and 1974, for the Oregon counties.
[See Appendix-B, Graph (i)-(ii)].
The reason for this may be that
fuel (and oil, etc.) and labor ratios vary little between different
types of husbandry processes.
One important possibility is that the fuel consumption and labor
required for the operation of a type of machinery (say a tractor)
may be relatively fixed (for a given terrain and soil
type, etc.).
Further, if the machines required for the crops are similar (i.e.,
tractor, combine, trucks for most of the small grain crops) and require
similar use rates, the above constancy of the fuel/labor ratio appears
plausible.
This collinearity between fuel and labor may lead to a
inulticollinearity problem in the estimation of the coefficients of
inputs.
The F-statistics used to test the weak separability
hypothesis depend upon the sum of the squared errors in two models,
multicollinearity may adversely affect this test.
This is true
particularly when collinearity is perfect and coefficient estimation
becomes infeasible due to singularity of the (X'X)-matrix.
The land quality index variable ought to measure the intrinsic
fertility of the soil.
not available.
However, such an index of land quality is
Often the indices available are derived from some
other "value of production" model, and their use in the present
context would be 'circular'.
Land quality indices for Oregon counties
are available, however, in dissertations by Holloway (1972) and
Thomas (1974).
The data on fertilizer input are also faced with the difficulty
of heterogeneous summation.
The census figures do not indicate
whether the liquid and dry fertilizers have the same nutrient content
(say for the primary component rt2).
There appears to be some
validity in assuming that the nitrogen concentrations in weight do
not vary too much between dry and liquid forms.
(This is based on
telephone conversation with Peter White, Pendleton Grain Growers
Cooperative, Pendleton, Oregon.)
Accordingly, the measure of
fertilizer input has been derived by adding the two types together.
Pesticide and herbicide inputs were not available in the Census
of Agriculture; though their effects are important they have been
left out in this study, resulting in some specification bias.
Precipitation was obtained for the chosen county weather station
and was regarded to be representative of the county situation.
The
temperature variable employed was not the average annual temperature.
A more relevant variable is believed to be the incidence (or lack)
of extremely cold temperatures.
A better index would consist of the
length of the period during which the lowest tolerable temperature
did occur.
Thus the relevant information would consist of the number
of days (in the growing season) when the temperature of the air and
ground did not fall below 46°F (the minimum tolerance temperature
for wheat.
This information was not readily available, although an
examination daily for each county for each year would have provided
this information.
The index chosen for temperature was, therefore,
the 'Number of days between dates with 32°F or below--a statistic
which is reported by Environmental Data Service.
It was felt that,
though this variable was not the ideal, it may still be explanatory
to some extent.
The irrigation variable used in this study represents a service
derived from a stock of infrastructure.
water applied in acre-feet.
serious drawback.
It does not measure the
This in itself is, however, not a
It is known in the case of irrigation that the
water application regime is as important as the amount of water
applied [Stegman (1980)].
The effect of a given water application
varies greatly depending upon the temperature, sunshine and wind
conditions and nature of the soil.
Thus, the errors in "input of
moisture" can vary greatly from one plot to the next, even under
uniform application of water.
The acreage of irrigated land (per
acre of total harvested acreage) has therefore some advantage in
that the measurement errors are small.
The assumption is that the
services from irrigation infrastructure definitely influence the
'biological process' itself, and so this variable captures some aspect
of "irrigation input."
The obvious disadvantage is that it ignores
the 'heterogeneity' within 'irrigated acres' category resulting from
water input differences.
The Units of the Variables
The units of the variables are as follows:
P,
precipitation, in inches per unit area;
1,
temperature index, number of days when minimum temperature is
above 32°F;
F,
Fertilizer per unit of land, tons/acre;
I,
irrigation service, percent of land irrigated;
N,
labor services, in thousands of 1954 dollars per acre, adjusted
on the basis of farms expenditure index for the USA.
[Prices Paid
by Farmers: Index Numbers, Annual Average, U.S., 1950-1977 (1g79)].
K,
capital services, in terms of fuel and oil expenses, in thousands
of i974 dollars per acre, adjusted as above;
Q,
land quality index, a pure number between a scale of 0-100
(or above)
'I',
wheat yield per unit of land, bushels per acre.
The data set is shown in Appendix F.
V.
RESULTS
Linear Computational Procedures
The translog model with linear restrictions was estimated using
the Time Series Processor (1976) package (called T.S.P.) from the
Computer Centre, The University of Western Ontario, Ontario, Canada.
The linear least squares estimation procedure used in the T.S.P.
package computes the estimates of the coefficients, their standard
deviation, the t-statistics, R2 and the Durbin-Watson statistics.
It
also gives the sum of the squared residuals, the standard error of
regression, the sum of residuals, F-statistics of the regression and
the estimate of the variance-covariance matrix of estimated coefficients
(see Appendix E for a sample).
The T.S.P. has a plotting capability, which allows for presenting
graphically observed and estimated dependent variable values, as well
as the errors.
The T.S.P. package also permits graphing the variables
against one another.
The T.S.P. package can be used for Farrar-Glauber
multicollinearity test, auto-correlation detection and correction.
has the capability to compute simultaneous system of equations.
It
The
other main advantage of T.S.P. lies in the fact that the matrix manipulations required in testing the linear restrictions can be readily
performed.
The nonlinear equation estimation procedure is also avail-
able within T.S.P. as will be discussed later on.
Model-A Results
As indicated in Chapter IV, the Model-A hypothesis is that the
weather variables and T are weakly separable from the other input
variables; F,I,K,N.
The weak separability assumed here is a simple
type with linear restrictions
= P,T and X
X.
= 0, where we have
= F,I,K,N.
The model has the following matrix form:
''FFFI'FK1FN
F
+
.P
o
+
T
+
I
K SN)
I
+ (FIKN)
'II1IK'IN
K
N
F
sym.
I
KKKN
K
1NN
N
The tests of separability, linear homogeneity and the Chow test for
regionally different production functions can be conducted by applying
various linear and nonlinear restrictions on the coefficients.
As
would be made clear consequently, the OLS procedure is adequate to
handle all of these restrictions except that of nonlinear weak
separabi 11 ty.
Regional Difference in Production Functions:
Chow Test Results
The difference in precipitation between eastern and western Oregon
results in major differences in the variety of wheat grown in these
areas.
west.
Winter wheat is commonly grown in the east, spring wheat in the
Therefore, it is expected that there are two distinct production
functions for these regions.
If linear-in-parameter TRANSLOG form is used to approximate the
production function, then the parameter vector,
in these two regions.
null hypothesis, H0
,
must be different
Thus the test involves the refutation of the
where
:
represent the parameters
and
for eastern and western Oregon, respectively.
This null hypothesis
can be put in the form of a linear restriction on the coefficients of
a more general model in TRANSLOG written as a simple linear-in-parameter
+
model, V = X
,
where X now represents all the linear terms
P,T,F,I,K,N, as well as the cross-products involving the last four.
Let the subscripts E and W represent eastern and western Oregon
observations, respectively.
The general model combining both the
eastern and western Oregon data will be represented as follows:
° J(E
IE1
0
IxE
I Ywi
XJ
w
The null hypothesis, H0
1+
IE1
I
is unit-matrix.
1=
G
2
tEwi
can be thought of as being a restric-
:
tion on the above model, such as, R
,
IEE1
1O,Vl
IEW
J
E
=
L
fEE
E
0, where R = [I
:
-I] and
The total sums of squares of the general
models above and of the partial models,
+
E
can be represented as e'e, eE'eE and e'e
E
and
respectively.
Yww+EE
Then under
the null hypothesis we can show that the ratio:
e'e-& e -e' e
EE
(e
eE + e
ww
eW)/(NE + Nw - 2k)
is the test statistic with F-distribution of (k, NE
+
of freedom.
tic.
- 2k) degrees
Thus the null hypothesis can be tested using this F-statis-
The results presented in the table on the following page indicate
88
Table 1
Chow Test Results
MODEL (1)
VAR1AbLS
y
MODEL (2)
+
=
=
X_3
+
MODEL (3)
''WWW
E
Oregon as
Eastern
a Whole
Oregon
Coeff. (t-values) Coeff. (t-vaiues)
w
Western
Oregon
Coeff. (t-values)
P
0.072
(1.34)
0.169
(2.84)
0.175
(1.34)
T
-0.032
(-.66)
-.006
(-.17)
-.443
(-1.35)
F
0.162
(0.49)
-.029
(-.05)
-.748
(-.69)
I
0.160
(0.79)
a.856
(3.34)
-.198
(-.35)
N
0.452
(0.58)
-1.101
(-.88)
0.749
(0.37)
K
-.640
(-.32)
1.461
(0.37)
-.203
(-.04)
F2
-.003
(-.18)
0.010
(0.39)
0.018
(0.35)
Fl
0.001
(0.07)
0.011
(0.69)
-.212
(-.48)
FrI
-.053
(-.80)
0.130
(1.29)
0.092
(0.63)
FK
0.044
(0.36)
-.177
(-.93)
-.298
(-1.12)
12
0.005
(1.11)
-.002
(-.64)
-.006
(-.23)
NI
-.042
(-1.25)
-.043
(-1.22)
-.042
(-.39)
KI
0.062
(0.87)
0.189
(2.38)
0.030
(0.14)
N2
-.119
(-1.20)
0.154
(0.86)
-.338
(-1.65)
0.382
(1.19)
-.556
(-1.04)
0.737
(1.21)
K2
-.273
(-.82)
0.406
(0.73)
-.284
(-.39)
Constant
3.578
(0.99)
5.578
(0.76)
5.423
(0.58)
R2
0.4116
0.7202
0.3790
13.71650
2.85246
7.99868
e'e
D.f.
142
71
54
N
159
88
71
e2
(:e2w
Computed F
2
+
CE)
/K
= 1.94
+
Table value of F(17125) at i= 0.05 = 1.65 and at(= .0i)= 2.03
CONCLUSION:
Reject the Null Hypothesis, H0: 3E = 34 at
o
.05.
(and at OL= .01, the test can be considered marginal).
that
and
are significantly different.
One fairly important
difference to be noted is that the coefficient of irrigation input in
the western region is not significant, though it is significant for the
eastern region at c = 0.01 level.
This perhaps reflects the fact that
in western Oregon, irrigation has only a minor impact due to the high
levels of precipitation.
In the east where precipitation is rather
low the precipitation coefficient is positive and significant (at
= 0.01).
However, there are no major differences in the significance
of T between the two regions.
Index variable for temperature does not
explain variance in wheat yield very well in either eastern or western
Oregon.
The strong multicollinearity between the capital and labor
services, K and N, may have caused the negative signs of N and K in
eastern and western Oregon, respectively.
The Fararr-Glauber test (Table 2) confirms the presence of multicollinearity in both the eastern and western regional models.
It
may be noted that multicollinearity makes suspect the sign, magnitude,
and variance of the coefficients.
Although the Chow test is valid
under multicollinearity, the power of the test would be reduced making
it less likely that separability hypotheses would be rejected.
Table 2 indicates the result of regressing each of independent
variable P,T,F,1,K,N with the others.
in the case of K,N.
The multicollinearity is severe
It may be noted that the Fararr-Glauber test is a
very cumbersome test when the number of variables are rather large.
For that reason, the test has been performed only with the set,
P,T,F,I,K,N rather than the set containing all of the square and crossproduct terms.
Table 2
Farrar-Glauber Analysis of Multicollinearity
(I)
Null Hypothesis:
Perfect Multicollinearity
H0: Det [X'X]
0
Test-statistics
610.756
Xdf15
Critical X2df=15 = 5.229 at
Conclusion:
(II)
c
0.01
Reject H0 at o= 0.01.
Regression of independent variables on others
H0: X.
is not linear in other X.'s.
Dependent variable
R2
F-statistic
Conclusion on H0
P
0.4935
29.82
Reject
T
0.3044
13.39
Reject
F
0.3210
14.47
Reject
1
0.2512
10.26
Reject
N
0.9557
659.67
Reject
K
0.9547
644.76
Reject
Conclusion:
The multicollinearity problem exists mainly between
labor N and capital service K.
The critical F with d.f. (15,153)
3.14
at c= 0.01 indicates that there exists a strong multicollinearity between N and K.
(III)
See Simple Correlation Coefficient matrix in Appendix-D.
91
Test for Cobb-Douglas Structure
The most restrictive of all separability restrictions that can
be applied to the TRANSLOG approximation is to reduce it from a logquadratic to a log-linear function.
The resulting Cobb-Douglas
function is strongly separable, i.e., the technology allows for pairwise separability of all inputs and so the elasticity of substitution
is unity between inputs.
The test involves deleting all second order
terms (generated from F,I,N, and K) in the regression and computing the
new sum of the squared errors e1e
F
N-K
to be used in F-statistics,
e'e
- 1).
q
e'e
Table 3 presents the results of the analysis for both eastern and
western Oregon.
Model (6) indicates that for western Oregon (at least) the CobbDouglas production function is a valid specification given the TRANSLOG
approximation.
The implication of this type of strong separability is
that weak separability between all F,I,N and K inputs holds and there-
fore, one need not test other linear and nonlinear weak separability
restrictions.
This was the reason for proceeding along with the nested
hypothesis sequence.
For the eastern Oregon, Model (5), however, the Cobb-Douglas
functional form is rejected, so the remaining possibilities are in
terms of weak separability amongst F,I,N,K or strong-pair-wise separability.
The latter does not imply Cobb-Douglas but merely additive
separability between F,I,N,K (i.e., all second order terms except the
Table 3
Test for Cobb-Douglas Structure
MODEL (5)
VARIABLES
MODEL (6)
MODEL (7)
Cobb-Doug] as
Cobb-Doug] as
Eastern Oregon
Coeff. (t-value)
Western Oregon
Coeff. (t-value)
Additively Separable
Eastern Oregon
Coeff. (t-value)
P
0.086
(1.46)
0.152
(1.32)
0.161
(2.57)
T
0.002
(0.06)
-.322
(-1.11)
-.002
(-.06)
F
0.227
(7.06)
0.227
(4.19)
0.104
(0.74)
I
-.012
(-1.09)
-.058
(-1.51)
0.448
(1.54)
N
- .028
(-. 42)
-.163
(-1.62)
-1.013
K
0.093
(0.65)
0.197
(1.00)
(-2.31)
5.406
(2.43)
F2
- .014
(- .84)
i2
0.005
(1.80)
N2
-.100 (-2.25)
K2
0.508
(2.41)
15.481
(3.21)
C
4.384
R2
0.5985
N-K
(8.78)
81
4.09310
q
10
e'e
2.85246
N-K
F = q-(
5.313
0.3126
64
8. 85271
0.6569
77
3. 49767
10
7.99868
6
2. 85246
e
-1) 3.09
Is
ee
Of (n1,n2)
Critical F
(10,81)
2.55 (c= .01)
d.f. (n1,n2)
(10,80)
Conclusion on
Rejected KID
Null Hypothesis
(10,64)
(6,77)
2.61 (= .01)
3.04 (= 0.01)
(10,65)
H0 Not Rejected
(6,80)
H0 Marginally
Rejected at
ot= .01
Rejected at
c= 0.05
cross-products are allowed).
The next step in the analysis consists
of performing these tests, in the order of additively restricted,
linearly restricted and nonlinearly restricted separabilities.
Table 3, Model (7) indicates the result of additive separability
for eastern Oregon.
This type of additive separability was also
rejected for eastern Oregon.
The implication is that there are some
interactions between F,I,N,K which are relevant in explaining the
variation of yield.
Test of Linear Homogeneity of TRANSLOG in F,I,N and K
Linear homogeneity of the TRANSLOG implies that we have
N
= 1.0
Ec
for the log-linear part of the function and
N
for all
0
in the log quadratic part.
i
=
The restriction equation, R
= r, can be
specified in terms of the following:
00111100000000000
0000001½ ½ ½ 0000000
R =0
00000000½ 00½ 01½ 00
0
0
0
0
0
0
½
0
0
1
½
000000000½ 00
andr=[l
0
0
0
0].
½
0
0
0
0
0½ 10
94
Table 4
Test for Linear Horr.ogeneity
MODEL (8)
-1
[R (x X)
=
-1
R ]
147.057
-1002.9
-725.723
7585.85
-1423.17
5494.90
9849.45
6948.37
7299.51
symmetrical
matrix
-1526.14
10546.6
7762.02
14033.1
14856.8
15890.2
=
F
(r-R
1.875
and
(i-P) =
0.185
,
r =
1
-.008
0.008
0
0.077
-.077
0
-.080
0.080
0
0.134
-.134
0
[R(X'X'R'](r-RB)
210.69
= 42.14
q.S
d.f. = (5,71)
(q.N-K).
F critical = 3.29 (at d.f. 5,70).
(ot
= .01)
Conclusion = Reject the null hypothesis H0:R
= r at
= 0.0
95
In this particular case, the most convenient
way to compute the
test statistics is to use the formula:
(r-RbY [R(X'X)R']
q
(r-Rb)
S2
The T.S.P. package allows this computation.
The results for the eastern
Oregon model are presented in Table 4, Model (8) below.
The matrix
[X'X]/S2 was first computed using the OLS procedure of the T.S.P.
package.
This was premultiplied by R' and postmultiplied by
R.
The
resulting matrix was then inverted to obtain
[R(X'XY1R'
S2
The coefficient vector b was also retrieved from
the OLS procedure in
the T.S.P. package.
Then (r-Rb) was computed from b and R and r.
The
linear homogeneity test was performed using the F-statistic.
Cr-Rb)' [R(X'XY
q
.
(r-Rb)
s2
The null hypothesis of linear homogeneity, or H0:R
the 1% level of significance.
= r, was rejected at
The model, it should be noted, already
presumes that total yield is a linear homogeneous function of
total
inputs such as land, capital, labor, fertilizers, etc.
The present test
of linear homogeneity (of the per unit land model)
is a test for an
additional restriction on the linear homogeneous technology
already
presumed for the 'totals' variable model.
Test of Linearly Restricted Weak Separability
Single Partitions
It was indicated in Chapter III, page 66, that hypotheses of linear
weak separability can be tested with single and double partitions.
There are seven single partitions possible with four variables F,I,N,K.
There are three partitions with pairs, [(F,I), (N,K)], [(F,N), (I,K)]
and [(F,K), (N,I)]; and four partitions with triplets, [F, (I,N,K)],
[(F,I,N),K], [(F,I,K),N] and [(F,K,N),I].
These models can be simply
estimated by deleting relevant cross-product terms from the quadratic,
TRAIl SLOG.
For example, the [(F,I), (N,K)] partition can be represented by deleting the FN, FK, KI, NI cross-product terms from the general quadratic
expression of TRANSLOG.
Single partition separability can be tested
using the F-statistic,
q
'
e'e
l' where e'e
represents the total
sum of squares for the model without the relevant cross-product terms
and e'e is that for the unrestricted TRANSLOG.
Table 5 and Table 6 below give the results of the restricted models
with single pair-wise and sing]
triplet partitions, respectively.
The
results indicate that amongst the single pair-wise partitions,
[(K,I),(F,N)] can be rejected, whereas, [(F,I),(N,K)] and [(K,I),(F,N)]
can not be rejected.
This implies that one of the four interactions
Fl, KI, FN, or KN is necessary to explain the variance of yield in the
model.
The results amongst single triplet partitions narrows the choice
from the four cross-product terms considerably.
Table 6 shows that
97
Table 5
Single Partitions (with pairs) on Eastern Oregon Model
MODEL (13)
Null Hypothesis
Variables
H:IF,i),(N,Kfl
Coeff.
(t-value)
MODEL (14)
H:[(F,N), (K,I)]
MODEL (15)
H:[(F,K) ,(N,I)]
Coeff. (t-value)
Coeff.
0.168
(2.69)
-.0006
(-.02)
(t-vaiue)
P
0.153
(2.67)
0.173
(2.91)
T
0.005
(0.16)
-.007
(-0.23)
F
0.171
(1.16)
0.207
(0.66)
-.340
(-. 52)
I
0.179
(3.14)
0.688
(3.56)
0.186
(2.09)
N
-1.325
(-1.24)
- .413
(- .65)
-1.297
(-2.06)
4.333
(1.66)
1.164
(0.40)
6.600
(1.94)
-0.015
(-.91)
-0.006
(-.39)
0.001
(0.04)
0.036
(2.69)
0.015
(0.26)
-.106
(-.72)
.002
(0.47)
0.032
(1.64)
K
2
Fl
FN
FK
0.003
(1.07)
-0.003
(-.94)
NI
KI
0.130
(3.36)
-0.046
(-.90)
-.140
(-2.20)
N2
0.033
(0.25)
NK
-.314
(-.76)
K2
0.562
(1.48)
0.072
(0.26)
0.662
(1.82)
11.904
(2.38)
6.861
(1.19)
17.2651
(2.64)
C
R2
e
d.f.
F
0.6879
0.7019
0. 6711
3.18513
3.03936
3.3 5281
88-13
(88-13)
(88-13)
2.0701
1. 1488
3. 1135
Critical F (4,70) at
Conclusion
on H0
= 3.01, (3.60), and at
Cant Reject
H0
Can't Reject
H0
= 0.05 (2.5)
Can't Reject H0
at
.01
Reject H0 at
= 0.05
Table 6
Single Partitions with Triplets on Eastern Oregon Model
Null
MODEL (9)
MODEL (10)
MODEL (11)
MODEL (12)
[F,(I,N,K)]
[(F,I,N,)K]
[(F,I,K),N]
[(F,K,N),I]
Hypothesis
(t_
(t_
(t_
Variables Coeff. Value) Coeff. Value) Coeff.
Coeff. (t-Value)
P
0.170
(2.85) 0.166
(2.73)
T
-.009
(-.27) 0.006
(0.17) -0.00004 (-.002) 0.00004 (0.001)
F
0.121
(0.86) 0.042 (0.13) -0.035
(-.06)
-.437
(-.64)
I
0.836
(3.79) 0.267 (2.87)
(3.12)
0.057
(1.74)
N
-0.435
K
0.169
(2.85)
0.617
0.162
(2.53)
(-.39) -.845(-1.35) -0.551
(-.87) -1.88
0.573
(0.19) 3.904
(1.44)
(0.60)
8.066
(2.19)
F2
-.006
(-.35) -.010
(-.64) -0.0005
(-.02)
0.005
(0.19)
El
-
-
0.019
(1.34)
-
FK
-
-
-0.050
(-.35)
-.191
(-.94)
EN
-
-
0.065
(0.64)
i2
-.003
0.005
(1.91)
MT
I,L
r1i
.UPJ.
KI
0.195
N2
-0.008
(-.06)-.083 (-1.6)
NK
-0.040
(-.10)
K2
0.016
(0.04)0.376
(1.47) 0.194
C
5.281
(0.95)11.89
(2.20) 8.36
R2
0.7095
0.6930
0.7090
0.6614
2.96114
3.12938
2.96661
3.45151
88-14
88-14
88-14
88-14
0.9017
2.2976
0.9470
4.9488
e'e
d.f.
F3
Value)
71
(-.87)
1
i
'\
J...JJ)
(3.13)
Conclusion
on H0
Can't reject
0.032 (2.38)
-
-
-.034 (-.57)
.00005(.O2)
n'
.V.J
r\
li
L..JUJ
-
-
-
-
Reject
2.131
-
-
-.003
(-.79)
-
-
0.101
-.056
(2.36)
(-1.48)
-
-
-
-
-
(-0.87) -0.36
(-.20)
-.327
(-.57)
(0.51)
0.98
(1.76)
(1.22)
19.286
(2.89)
-
-
Can't reject
Reject
[(F,N,K),I] and [(F,I,N)J are rejected.
the KI cross-product term is deleted.
In both of these partitions,
On the other hand, [(I,N,K),F]
and [(I,F,K),N] are not rejected indicating that deleting the EN and KN
cross-product terms is permissible.
Therefore, it appears that the
cross-product term KI cannot be deleted.
Though KI appears important,
Fl can also explain the variance of yield to some extent as indicated by
non-rejection of [(F,I),(N,K)] partition.
This generates some ambiguity
as to whether the separation of K and I alone are responsible for the
rejection of [(F,N,K),I], [(F,I,N),K], [(F,N,K),IJ and [(I,F,N),K].
Thus double partitions were considered necessary to resolve this
ambiguity.
The ambiguity would be removed completely if partitions of
the type [(F,N),K,I], [(F,K),N,I] and [(N,t),F,K] alone were to be
rejected.
Thus, if [(N,K),F,I] and [(K,I),F,N] were not rejected, this
would imply that the Fl cross-product term was not necessary to explain
the variance in yield.
In this case alone the ambiguity can be reduced
in the importance of models with KI vis-a-vis those with Fl.
Double Partitions
In case of four variables, F,I,N and K, there are altogether six
double partitions.
These always contain a pair, e.g., [(F,I),N,K],
[(N,K),F,I], [(F,N),K,I], [(K,I),F,N], [(F,K),N,I] and [(N,I),F,K].
These models are more restrictive than single partitions, since more
cross-product terms are deleted.
remains unchange
before.
,
The estimation procedure, however,
and the F-statistics can be computed exactly as
100
Table 7
Double Partitions
Null Hypothesis
Variables
MODEL (16)
H0:[(F,I),N,K)
MODEL (17)
H0:[(N,K),F,I)]
(t-value) Coeff.
Coeff.
(t.value)
MODEL (18)
H0:[(F,N)K,I)]
Coeff.
(t-value)
P
0.159
(2.62)
0.163
(2.57)
0.162
(2.55)
T
0.007
(0.20)
-0.003
(-.08)
-0.002
(-.06)
F
0.202
(1.43)
0.088
(0.59)
0.054
(0.16)
I
0.167
(3.06)
0.048
(1.56)
0.045
(1. 54)
N
-0.591
(-1.31)
-1.346
(-1.21)
-1.09-
(-1.70)
K
3.387
(1.49)
5.865
(2.22)
5.670
(2.06)
F2
-0.011
(-0.70)
-0.015
(-.90)
-0.014
(- .84)
Fl
0.035
(2.61)
-
-0.010
(-.17)
FN
-
FK
i2
0.002
(0.92)
0.005
(1.80)
0.005
(1.79)
-0.602
(-1.32)
-0.060
(-.45)
-0.104
(-2.02)
-
-0.139
(-.33)
-
NI
KI
N2
NK
K2
0.325
(1.51)
0.617
(1.56)
0.533
(2.04)
11. 297
(2.30)
15.839
(3.18)
15.890
(2.92)
R2
0.6852
0.6574
0.6570
e'e
3.20947
3.49281
3.49639
d.f.
88-12
88-12
88-12
1.7653
3.1747
3.1925
F(d.f. = 5,71)
Critical F (c= 0.01)
Conclusion
= 3.25
Can't Reject
H0
F (c.= 0.05) = 2.33 at d.f. (5,70)
Reject H0
Reject H0
101
Table 7 (Cont.)
Double Partitions
Null Hypothesis
Variables
MODEL (19)
MODEL (20)
MODEL (21)
H0:[(K,I)F,N)
H0:[(F,K),N,I)
H0:[(N,I),F,K)
Coeffi. (t-value) Coeff.
(t-value) Coeff.
(t-value)
P
0.173
(2.93)
0.160
(2.54)
0.169
(2.72)
T
-.007
(-.22)
0.0005
(.01)
-.003
(-.09)
F
0.132
(0.99)
-0.375
(-.56)
0.121
(0.86)
I
0.682
(3.57)
0.048
(1.64)
0.183
(2.07)
N
-0.533
(-1.22)
-1.35
(-2.13)
-0.972
(-2.24)
K
1.595
(0.67)
7.30
(2.14)
4.771
(2.14)
-0.006
(-.41)
-.0005
(-.02)
-0.012
(-0.72)
F2
Fl
FK
i2
NI
-
-
-
EN-
-
-
-
-
-0.003
(-.95)
-
-
-
-0.111
(-0.74)
0.005
(1.84)
-
-
KI
0.128
(3.37)
-
N2
0.053
(1.20)
-.134
-
NK-
(-2.09
-
-
-
-
-
-
-
0.001
(0.43)
0.032
(1.67)
-
-
-0.107
(-2.43)
-
-
K2
0.113
(0.49)
0.728
(2.00)
0.450
(2.12)
C
7.550
(1.48)
18.727
(2.86)
14.1328
(2.92)
R2
0.7016
0.6593
0.6689
e'e
3.04214
3.47284
3.37578
d.f.
88-12
88-12
88-12
0.9329
3.0754
2.5926
F(5,71)
Critical F (o
Conclusion
0.01) = 3.25; F (OL= 0.05)
Can't Reject
H0
Reject H0
2.33 at d.f. (5,70)
Reject H0 at
c
0.05
102
The results are indicated in Table 7 below.
the results obtained with single partitions.
These again support
The double partitions,
[(N,K),F,IJ, [(F,N),K,IJ, [(F,K),N,IJ and C(N,I),F,KJ are again strongly
rejected.
However, [(K,I),F,N] cannot be rejected while [(F,I),N,K] is
marginal.
Thus the ambiguity between Fl and KI cross-products is not
entirely resolved.
But because F-statistic for Model (19) with KI is
much smaller than that for Model (16) with Fl, Model (19) with Ki
appears to be the preferred model.
It may, however, be noted that the significance of the Fl term,
given that the model already contains the KI term, is rather low.
This
type of significance can be tested by considering the models with the
following partitions, [(I,F,F),N] and [(K,I),F,N].
Here the model with
the partition [(I,F,K),N] is regarded as the unconstrained model and the
[(K,I),F,N] partition represents the restricted model.
was computed, using F =
of freedom.
N-K
ee
-1), to be 0.95 with (2,74) degrees
The critical value of F2
= 0.05 (Table 8).
The F-statistic
74
is 490 at c0.0l and 3.12 at
Consequently, the cross-product terms Fl and FN can
be deleted from the model with partition [(I,F,K),N], assuming this
partition is correct.
The significance of the KI term given that the model already
includes the Fl term is, however, not very strong.
It can be seen that
with the partition [(F,I),K,N] the F-statistic is 303.
this result is marginal, and rejection may be assumed.
At c
0.05,
The results
still involve the ambiguity between the models with KI and Fl terms.
It
may also be noted the test of significance of Fl given KI in the model
103
Table 8
Conditional Double Partitions
MODEL (21)
Maintained
Hypothesis
MODEL (22)
Model Hypothesis
H0:[(F,I,K),N]
Terms
MODEL (23)
Model Hypothesis
H0:[(K,I),F,N]
H0:[(F,I),K,N]
ee
2.96661
3.04214
3.20943
R2
0.7090
0.7016
0.6852
K
14
12
12
N
88
88
88
-
0.95
3.03
-
4.90
4.90
-
3.12
3.12
-
Can't Reject
Can't Reject
atc= 0.01 or
atc= 0.01
e
e
F =
ee
Critical F274
-1
at= 0.01
Critical F274
atc,= 0.05
Conclusion
on H0
oi..
= 0.05
Marginal Rejection
at= 0.05
104
is very low.
A similar test of KI given Fl in the model indicates that
KI is significant.
Thus, even though the results are still ambiguous,
inseparability between K and I appears to be more likely than that
between F and I.
The choice of separability between F and I versus K
and I must be resolved in some other way, i.e., through the imposition
of further restrictions on the model.
This involves nonlinear restric-
tions and Sadan type models.
Mitigation of Multicollinearity
The Cobb-Douglas structure for western Oregon results in negative
values of the coefficients for T, I and N.
It may, however, be noted
that all the coefficients except that of fertilizer are insignificant
at cx = 0.01.
This is caused by the strong multicollinearity between
K and N, K2, N2, etc.
The inflationary effect on the variance of the
estimated coefficients due to multicollinearity has resulted in impre-
cision in the estimated coefficients, and possibly in improper signs
of some of the coefficients.
The Census data for 1954-1974 period indicates that the K/N ratio
has remained relatively stable over this period (See Graph 1
in Appen-
dix B), implying that labor services tend to be proportional to capital
services for aggregate county-level data.
In the case of the eastern Oregon linearly restricted and unrestricted TRANSLOG functions, the precipitation and irrigation variables
P and I are almost always significant.
The restrictions have little
effect on the sign and the magnitude of the coefficient of P.
The sign
and magnitude of the coefficient of I is not so stable, however, but
105
the sign and magnitude remain more or less the same when the restriction includes either the KI or the Fl cross-product term in the model.
The sign and magnitude of the coefficient of irrigation service, I,
remains areound 0.20 in various single and multiple partition models
where Fl occurs without KI.
The coefficient is around 0.60 in models
where KI occurs without Fl.
The double partition model [(K,I),F,N], has only three significant
variables, P,I, and KI.
Again there exists multicollinearity amongst
the N,K,N2 and K2 terms which magnify the variance of the estimated
coefficients.
There are various alternative methods that may be
employed to resolve this problem as indicated above in Chapter II, page
36.
a)
Models with Fixed K/N Ratio
Amongst the various alternatives available, such as the ridge
estimators, principal component, and mixed estimate approach, the
latter appears to be suitable here.
The ridge estimators are disadvan-
tageous in the present situation because of the diverse values of the
coefficients.
The principal component approach, on the other hand, is
not attractive because the estimated components are difficult to
interpret (although intuitively, the separable parts of the production
function should correspond with the major components, one cannot impose
the separability a priori; consequently, this approach is not feasible).
The relative stability of the K/N ratio therefore indicates the
mixed estimators as a natural choice in mitigating problems of multicollinearity.
The implication, in practical terms, is that one of the
106
K or N terms can be deleted from the estimation procedure.
The inter-
pretation of the value of the coefficient of the remaining variable,
however, is changed.
The coefficient now represents the effects of
both the variables K and N.
For example, if we assume that K/N ratio
is fixed, we can delete N from the regression equation.
The coeffi-
cients of all K, KI and K2 terms then represent the joint effects of
K and N, and their individual effects cannot be separated.
It was also noted that the temperature index, T, was not significant for all of the regressions.
often close to zero.
The value of the coefficient was
These facts were considered in the following
regression runs shown in Table 9.
in favor of capital service, K.
Labor service, N, has been deleted
Though all the previous runs with
single and double partitions could have been duplicated, this was
felt to be unnecessary since the results of the four models presented
here are not much different from the earlier ones.
Model (24) with the assumption of fixed K/N ratio, has an F-
statistic of 065l8 when the full model (2) is constrained with fixed
K/N ratio and deleting T.
This indicates that the fixed K/N ratio
model cannot be rejected at a = 00l.
The main effect of deleting
T and N appears to be a slight increase in the precision of the
remaining coefficients.
The coefficient of capital service, K, is
still not significant at a = 0.05, however.
The conclusions of the
previous runs with both I and N included, does not change even after
deletion of T and N.
Table 9 also indicates that the ambiguity
between models with Fl and KI cross-product terms still persists.
107
Table 9
Models without I and with Fixed K/N Ratio Represented by K
MODEL (24)
H0:"/(F,I,K)
Variables
MODEL (25)
MODEL (26)
MODEL (27)
H0:P/[(F,I),KJ H0:P/[(F,K),IJ F-10:PJ[(K,I),F
Cceff. (t-value) Cneff. (t-value)Coeff. (t -v1ue)Coeff. Ct-value)
P
0.158
(2.87)
0.136
(2.47)
0.129 (2.15)
0.148
(2.75)
F
0.384
(0.39)
0.301
(2.63)
0.718 (1.71)
0.205
(1.84)
I
0.664
(3.55)
0.187
(3.78)
0.045 (1.54)
0.760
(4.27)
K
-.733
(-.60)
0.880
(0.84)
0.474 (0.40)
-.991
(-.88)
F2
-.006
(-.30)
-.001
(-.07)
-.015 (-.68)
0.003
(0.21)
i2
-.003
(-.97)
0.002
(0.97)
0.005 (2.04)
-.003
(-1.07)
-.111
(-.80)
0.091
(0.88)
0.003 (0.03)
-.134
(-1.16)
Fl
0.021
(1.52)
0.040
(3.31)
-
-
FK
0.038
(0.39)
-
-
-
-
KI
0.110
(2.69)
-
-
C
3.042
(1.05)
6.355
K2
R2
e'e
.
0.7060
0.6775
0.117 (1.12)
-
6.145 (2.13)
0.143
(4.03)
2.29
(0.81)
0.6386
0.6955
2.99734
3.28729
3.68462
3.10418
F1n2)
0.6518
1.4672
2.8079
0.8494
(n1,n2)
(6,77)
(8,77)
(8,77)
(8,77)
2.74
2.74
2.74
(8,80)
(8,80)
(8,80)
Critical-F (ct= .01) 3.04
(n1,n2)
Conclusion
at
(2.35)
-
-
CL level
0.01
(6,80)
Can't Reject
H0
Can't Reject
H
Reject H0
Can't Reject
H0
The results are not dramatically different when we delete I and N
variables, probably because K2 is still highly correlated with K,F2
with F and i2 with I.
The results could improve if the square terms
F2, K2 and K2 are deleted.
The deletion of F2 only appears to be per-
missible because its coefficient is small and insignificant at
= O0l in all equations of models (24), (25), (26) and (27).
b)
Models with only KN Term
Yet another approach to reduce multicollinearity problems is as
follows.
The variables K and N, and their squares, were eliminated and
only the cross-product term KN preserved.
This approach emphasizes the
possible nonlinear nature of the effect of K and N inputs upon yield.
The multicollinearity is smaller between the rest of the variables as
indicated by the correlation coefficients.
The results are indicated
in Table 10 below.
The models tested are again limited to those with Ft and KI
variables.
Results similar to the previous ones are obtained.
(30) with KI does better than Model (29) with El, as before.
Model
The
variable T is insignificant, so could have been eliminated without
changing the results.
The coefficients for other variables P, F, I,
are significant in all three models.
But the values of the coefficients
of I are rather unstable, though the coefficients for P and F are
stable.
This type of pattern was also exhibited in the constant
ratio model discussed previously.
The coefficient for NK is small but negative and significant at
= 005 in Model (30) with KI term.
It is positive and not significant
109
Table 10
Models with Cross-product Term KN
MODEL (28)
Null Hypotheses
Variables
HO:SFIKI
Coeff.
MODEL (29)
0
MODEL (30)
0
HO:3FI
0
HO:3K1
(t-values)Coeff. (t-values) Coeff. Ct-values)
P
0.159
(2.94)
0.130
(2.37)
0.161
(2.90)
T
-.006
(-.18)
0.019
(0.59)
-.022
(-.69)
F
.299
(2.78)
0.280
(2.52)
0.299
(2.70)
I
.545
(3.67)
0.190
(3.82)
0.572
(3.76)
KN
-.010
(-1.47)
0.001
(0.26)
-.015
(-2.13)
F2
.002
(0.17)
-0.002
(-.16)
0.009
'(0.66)
-.002
(-.60)
0.003
(1.17)
-0.001
(-.45)
Fl
.030
(2.35)
0.041
(3.34)
KI
.080
(2.53)
4.409
(15.22)
C
-
4.088
(15.19)
-
-
0.105
(3.48)
4.509
(15.30)
R2
0.6999
0.6753
0.6787
e'e
3.05933
3.30991
3.27552
F
0.7356
1.4233
1.3163
7,71
8,71
8,71
d.f.(n1,n2)
Critical F
(c,.=
d.f. (n1,n2)
.01)
(2.91)c,= 0.05 (2.14) ot
(7,70)
(8,70)
0.05 (2.07) c,= 0.01 (2.77)
(8,70)
110
at a = .05 in Model (29) with the Fl term included, and negative and
marginally significant at a = 0.05 in models with both Fl and KI.
The
coefficient for i2 is generally not significant in all three models,
the value is small, negative in Model (30) with KI and positive in
Model (29) with Fl.
Model (28) with both Fl and KI term has the lowest F-statistic
and the highest R2.
Coefficients of both Fl and KI terms are positive
and significant at a = 0.01.
But T. F2. i2 and NK have coefficients
that are insignificant at a
0.01 level.
are negative.
The coefficients T, i2, NK
The next best model appears to be the Model (30) with KI.
The coefficients of T, F2, r2 terms are not significant at a
0.05.
The parameters are nearly equal in Model (28) with only KI term.
This
indicates that models with only the KI term are more acceptable than
those containing Fl.
c)
Models with N or N2 and K or K2 Only
Yet other models were tested using combinations of N, K, N2 and
K2
terms.
The deletion of N2 and K2 was expected to reduce multi-
collinearity to some extent.
Again the tests were limited to the two
competing models, one with the Fl term and the other with the KI term.
The results indicate that models with KI do consistently better than
the models with Fl.
The coefficients of T, F2, K2, N2 or N are
insignificant at a = 0.01 in all six models.
The models with KI have
significant coefficients for K or K2 but not for N2 and N, at a = 0.01.
Amongst models (32), (35) and (36), the model with the smallest
sum of squared errors is Model (35) (with K2 instead of K), and here
111
Table 11
Models with Single Terms with N and K-variables
MODEL (31)
MODEL (32)
HO:B2=3KI3KIO
HO:SN2=K2=FI=O
FI
Variables
O
MODEL (33)
HO:N2K=3KIO
KI
Coeff. (t-values)
Coeff.
FI
(t-values) Coeff.
(t-values)
p
0.134
(2.35)
0.144
(2.58)
0.135
(2.38)
T
0.163
(.50)
-0.011
(-0.35)
0.015
(0.46)
F
0.296
(2.39)
0.205
(1.71)
0.301
(2.44)
I
0.193
(3.78)
0.672
(4.22)
0.194
(3.81)
N or N2
0.011
(0.17)
-0.029
(-0.48)
0.014
(0.24)
K or K2
-0.051
(-0.35)
0.364
(2.33)
F2
-0.001
(-0.08)
0.002
(0.13) -0.001
i2
0.003
(1.12)
-0.003
Fl
0.041
(3.30)
KI
C
(
3.956
)
(8.43)
0.006* (0.44)
(-0.06)
(-0.87)
0.003
(1.10)
-
0.042
(3.33)
*
0.126
(3.95)
5.642
(9.73)
-
-
4.086 (13.27)
R2
0.6757
0.6920
0.6760
e
3.30595
3.13975
3.30298
Fnln2
1.6125
1.0216
1.6020
(n1n2)
(7,71)
(7,71)
(7,71)
Critical F
(o= .01)
(n1,n2)
Conclusion
Can't Reject
H0
*
means K' instead of K.
Can't Reject
H0
Can't Reject
H0
112
Table 11 (Cont.)
MODEL (34)
H0 :3K2=3N=3KI=O
MODEL (35)
H0 .3K'3N2H3FIO
Fl
Variables
Coeff.
MODEL (36)
H0 :N3K23FI=O
0
KI
(t-values) Coeff.
0
KI
(t-values) Coeff. (t-values)
P
0.134
(2.35)
0.145
(2.62)
0.144
(2.56)
T
0.017
(0.51)
-0.009
(-0.29)
-0.014
(-0.43)
F
0.294
(2.43)
0.200
(1.67)
0.221
(1.88)
I
0.193
(3.79)
0.705
(4.30)
0.670
(4.20)
0.0004*
(0.07)
0.318
(2.18)
N or N2
-0.001
(_0.18)*
-0.028
K or K2
-0.049
(-0.37)
-0.038
(-0.48)
(_2.44)*
F2
-0.001
(-0.09)
0.002
(0.12)
0.003
(0.21)
i2
0.003
(1.13)
-0.003
(-0.96)
-0.003
(-0.88)
Fl
0.041
(3.31)
-
KI
-
-
3.933
C
(7.43)
-
-
-
0.132
(4.04)
0.125
(3.93)
4.750
(13.80)
5.580
(9.05)
R2
0.6757
0.6940
0.6911
ee
F12
3.30586
3.11923
3.14886
1.6122
0.9486
1.0539
(7,71)
(7,71)
(7,71)
(n1,n2)
Critical
F (c.=.01)
(n1,n2)
Conclusion
Can't Reject H0
Can't Reject H0
*
represents N2 or K2 as the case may be.
Can't Reject H0
113
the coefficient of K2 is negative and significant at
= 0.05.
However, models (2) and (6) (with K and N) have positive and significant
(at c
0.05) coefficients for K.
preferred to Model (35).
The latter two models may thus be
In all six models, coefficients of P, F,
and the constant C are positive and significant at
Nonlinear Computational Procedure
I
= 0.01.
-
TRANSLOG models with nonlinear restrictions were estimated using
the Time Series Processor (TSP) package.
The nonlinear least squares
estimation procedure used in TSP, or any other package, must utilize
some kind of numerical iterative procedure.
There are many types of
numerical methods available which can be used to minimize the sum of
squares (or some such objective criterion) to determine the nonlinear
least squares estimates.
What is required is that the functional form
be known explicitly.
The diagram below indicates the procedure.
initial parameter vector
Starting with an
a step length, t, is chosen.
The next
value of the parameter vector is given according to the vector equation,
n+1
n
where
+
parameter
is the nth approximation of the true
and ç.is the step direction vector chosen at the
th
iterative step.
Unless a termination criteria is fulfilled at the nth
step, a new
is generated and a new
n+l
n+2
computed, and so on.
In the TSP the gradient approach is taken to decide the new step
direction, whereas the step length is determined on the basis of
specified tolerance limit.
The new s-vector must not exceed the
a
114
tolerance limit plus or minus the previous s-vector, for convergence
to be achieved; and at the same time, the sum of the squared errors
must diminish (or stay the same).
Let us define the sum of the squared error terms as S(s) = e'e
[Y - f (,X)]' [Y - f (,X)]; then an appropriate step length, t, and
direction, E, is such that:
(3+ t)< S (3).
The direction vector
is such that S
+ t
(
tion of t, where t is a small scaler.
is a decreasing func-
)
Thus we must have,
< 0,
t=0
i.e.,
+ t
dS(
dt
d(n+t)
1
1sJ
)I
{s
I
3nj
is negative.
I
dt
I
ni.
Thus the gradient
of the sum of the squared
errors. S, may be utilized to define the new step direction vector,
as follows:
=
where P
Pfl[Tj
is any positive definite matrix such that
1s
1
J>
tnI n1Ij
0.
This assures us as required that,
dS
aT
1i
1
11
J'
t31nl
Now the question arises as to the choice of
fSI
1
n8kj <
0,
115
Figure 11:
Iterative Nonlinear Least Squares Algorithms
116
The Gauss-Newton method involves approximating a positive definite
with the Hessian of S.
Since, S(s) = (e'e) = (y-f(,X))(y-f3,X))
the Hessian of S(3) is given as,
H(S(3)) = H(3) = Z(3)'Z()
where, Z(3) =
-
(y -f(,X )
Assuming that we are near the true
.
value, the second term on the right [(yt_f(3,Xt) is close to zero]
can be ignored.
Thus the choice of the positive definite matrix
is provided by,
Z() Z()]*
P
Therefore,
S(
3n+in+tn+tPn)[i3j
And for a unit step length, t
1, we get,
+ [Z()' Z(3)]1
1
L'3nJ
or,
+ [Z(3 ) 'Z(3)]
n+i
n
.
n
Z($)' [y-f(3 ,X)]
n
n
or,
n+1
[Z
) z(]
Z(
n
[y-f(3X) + Z(3
nn
(The last expression simplifies to the last but one, after some multiplication and expansion).
Thus
model,
may be thought of as being the OLS estimator of the
= Z(3). 3 +
,
where,
'
= y-f(3,X) + Z(3) 3
.
This can
be recognized as the linear pseudo model at 3n' approximating the nonlinear model, y=f(3,X)
The
with
(instead of the true parameter 3).
Gauss-Newton algorithm can be interpreted as a sequence of OLS
estimations.
In each step of computing 3,., one computes the least
squares estimator for a linear approximation (based on the previous
117
of course) of the true non-linear model [Judge et al (1980), pp.
735].
The Z(3)Z( 3) matrix should be non-singular in general, so that,
the new
can be computed.
This, however, may not always be true;
Z (3)Z(3 ) can be singular and therefore not positive-definite.
In
such a situation, convergence may be rather slow and this should be
considered in estimating large nonlinear models.
Model-A Nonlinear Restrictions
It has already been indicated the nonlinear weak separability
restriction allows one to write the TRANSLOG function as a contracted
polynomial, i.e., one can express the quadratic part of the TRANSLOG
as a quadratic in linear functions (for each separable group).
The resulting contracted quadratic has the following form,
y
AlA
+o(O x) + Y11 (ox) 2
+
The oB and
IN
(0 x)
.
2
AI"
(0 x).
are linear coefficient vectors, which also gene2
rate the quadratic tenns (B X)
ing
A
eX) + YNN(O X)
2
,
(B X)
I
and (OX) .(O X).
Substitut-
1eX = f(X),cNOX = g(X), we have,
A
+ f(X) + g(X) + a
X) + b f (x)
.
A
A
2
g(X) + c g
(X)
which indicates a contracted quadratic in f and g or the
-function,
under the non-linear restrictions of weak separability.
Within Model-A, F,I,N and K can possibly generate as many nonlinear separability partitions as was enumerated with linear separability partitions.
The iterative estimations required in nonlinearly
restricted TRANSLOG however indicates that an exhaustive analysis
118
analogous to the linear restrictions would not be appropriate.
Thus
to simplify the computational complexities and costs, it was necessary
to reduce the number of variables as well as the number of restrictions
to be evaluated.
The tolerance for convergence was maintained at
0.005.
The temperature index, T, was deleted since it was not significant
at all in all previouscases.
The number of variables was also reduced
in the F, I, N, K group by assuming that (K/N) ratio was fixed and
so (N,K) could well be represented by K.
Thus the model chosen, on
which to test the nonlinear restrictions on, was TRANSLOG with estimated form:
=
3P
+
B
KK
+
BEE
K2
+
El
+
F.I
BKK
+311.12
+ BFF.F
+ 3FK F.K + 3K1 KI
Table 12 indicates the results.
Model (37) was estimated using
the expression,
y
= a +
a.P
+ aF.F + a1.I
+ b (aF.F + a1.I)2 + C K2
+ d (aFF + a11) K
and models (38) and (39) were also estimated using similar expressions.
Model (37) can be rejected because assymptotic F-statistic of 7.7812
implies the rejection of H0: [(E,I),K] at c
=
The result
0.01.
indicates that P and F coefficients are significant at c = 0.01
The coefficients of K,I, (aFF+arl)2 and (aFF+AII)K are not significant
at ot= 0.05.
Model (38) has P,I and F coefficients that are significant at
=
0.05,
=
0.01 and
=
9.10, respectively.
The coefficents of
(aFF+aKK)2, i2 and (aFF+aKK) are insignificant at ci.
=
0.05.
119
Table 12
Non-linear Weak Separability Restrictions
TRANSLOG with K/N ratio
fixed and I deleted
Cobb-Douglas Process Functions
MODEL (24)
Null
MODEL (37)
H :P/[F,I,K]
MODEL (39)
H :[(F,I),K]/P H :[(F,K)I]JP H :[(K,I),F]JP
0
Hypothesis
(t-
Variables
MODEL (38)
0
(t-
Coeff. values)
Terms
(t-
(t-
Coeff. values) Coeff. value) Coeff. value)
P
0.158
(2.87)
P
0.12
(2.00) 0.12
(2.17) 0.12 (2.51)
F
0.384
(0.39)
F
0.88
(2.10) 0.24
(1.64) 0.50 (1.98)
I
0.664
(3.55)
I
-.002 (-.03) 0.24
(2.70) 0.20 (3.74)
K
-.733
(-.60)
K
1.03
(0.75) 0.39 (0.92)
F2
-.006
(-.30)
(aFF+aII)2
-.02 (-0.90)
-
-
i2
-.003
(-.90)
(aFF+aII)K
0.17
(3.48)
-
-
K2
-.111
(-.80)
0.05
(0.36)
-
-
Fl
0.021
(1.50)
(aFF+aKK)2
-
-.06
(-0.22)
-
FK
0.038
(0.40)
(aFF+aKK)I
-
0.16
(1.55)
-
-
0.002
(0.72)
-
KI
C
0.110
3.042
K2
2
(2.69)
(1.05)
(0.90) 0.07
(aII+aKK)2
-
0.06 (1.37)
(aII+aKK)F
-
0.16 (2.38)
-
0.01 (-.70)
F2
C
-
7.82
(2.8)
4.54
(8.53) 5.76 (3.62)
R2
0.7060
0.6169
0.6753
0.6787
e'e
2.99734
3.90603
3.30971
3.27528
Fn1,n2
7.7812
2.6749
2.3800
(n1,n2)
(3,77)
(3,77)
(3,77)
4.04
4.04
4.04
(3,80)
(3,80)
(3,80)
Critical-F
(n1,n2)
= 0.01
Reject H0
Can't Reject
H0
Can't Reject H0
120
Using the F-statistics of 2.67 one cannot reject the H0:[(F,K),I) at
Model (39) on the other hand has P, F and I coefficients
= 0.01.
significant at c
= 0.01, a = 0.05, and a = 0.01, respectively.
However, the coefficients of (aII+aKK)2 and F2 are not significant
even at a= 0.05; the coefficient of (aII+aKK)F is significant at
c
= 0.05.
The coefficient of K, though positive, is not significant
at o= 0.05.
Using the assymptotic F-statistic of 2.38, the
null hypothesis H0 = [(K,I),F] cannot be rejected.
In conclusion, there appears to be some ambiguity amongst
weak nonlinear separability between (K,I) and F and that between (K,F)
and I.
There is, indeed, very little difference between these two
models from the standpoint of
or e'e.
But the [(K,1),F] model does
marginally better and the coefficient of the cross-product term
(aJI+aKK)F is significant at a
0.01.
Quadratic, Cubic and Quartic Approximation to Sadan Model
The results from linear and nonlinear weak separability restrictions indicate that a model with the KI term provides the best explanation of the data.
Though this conclusion appears reasonable, there is
still some ambiguity.
In the case of the linear restriction models,
those with Fl terms are strong contenders.
In the case of the non-
linear restriction models, the model with the FK term is the closest
contender.
This ambiguity can be resolved either by utilizing more
data or by imposing more restrictions on the production function.
The latter approach is methodologically more appropriate because
121
it can deal with ambiguous situations that may persist in spite of the
data set extensions.
Sadan's thesis of perfect process complemen-
tarity provides us with a set of further restrictions.
As discussed in Chapter III, though the Sadan Model
is discon-
tinuous in its parts f and g, it may be approximated through the use
of upside down parabolic cylinders or simple saddle surfaces in the
y-f-g space.
When f and g are linear in variables, the parabolic
cylinder, y =
a
+ f + g +
n
(f-g)2, becomes a quadratic in F, I and K.
When f and g are themselves quadratic because of KI or Fl terms,
the parabolic cylinder y
F,
I and K.
= a
+ f
g
+ ii
(f-g)2 becomes a quartic in
When the exponent of (f-g) is 1.5, the expression for y
in terms F, I and K becomes a cubic.
Alternatively, the cubic of
simplest form in F, I and K can also be equivalent to a saddle surface
y =
a + f
+ g + rfg.
Though it appears that the dimension of the
production function has increased (to a cubic or quartic from quadratic),
the functional form is more constrained due to the specific parabolic
or saddle surface specified in the y-f-g space.
In the case of the inverted parabolic surface y =
a + f + g +
the coefficient n should be negative to support Sadan's thesis.
In the case of the saddle surface, y
= a
efficient should be positive and small.
+ f + g + nfg, the n co-
Both of these surfaces impli-
citly imply that there is an optimum process-mix such that yield y is
maximized along the 45° line in f-g plane.
Though the 45° line optimum
focus is not necessary, it makes it convenient to compare and interpret
the results.
122
Quadratic Approximations to the Sadan Model
The quadratic approximations to the Sadan Model is tested using
linear separable parts in F, I and K.
constant as before.
The (K/N) ratio is assumed
Models (40) and (41) represent models with separ-
ability [(K,I),F] and [(F,K),I), respectively.
These two models are
chosen because the test of nonlinear separabilities were not rejected
in these cases.
For [(F,I),K] and [(K,I),F], linear weak separability
was not rejected, and so Model (42) tests for separable parts (aKK) and
(aFF + a11) in a Sadan model.
However, as in Models (40) and (41), the
optimal process-mix is constrained along a 45° line in the f-g plane.
Assuming that the general unconstrained Model (40) in Table 13 is
correct, the assymptotic F-statistics were computed for these cylindrical surfaces in y-f-g space, using the formula,
y = a + f + g +
(f-g)2.
The results of Table 13 indicate that Sadan type perfect complementary process functions which are linear in F,I or K are incompatible
with the data.
The rejection can occur because of (i) the linear
process-functions imply strong separability between the grouped inputs,
and no such strong separability actually exists, or (ii) because the
process-complementarity is not as high as is required by the Sadan
model.
Because of this, the present results cannot be taken as a re-
jection of the Sadan type model.
Actually, since strong/pairwise
separabilities have been rejected before, the present result is likely
to be caused by assuming linear process-functions, f and
g.
The significant parameter in Model (40), (41) and (42) in Table 13
123
Table 13
Inverted Cylinder, y = a + f + g + ri
(f-g)
(Quadratic Approximation to Sadan Model)
MODEL (40)
Hypothesis
o
Variable
Parameters
MODEL (41)
f = aK.K+aI.I
f
g=a.F
g=a11
Coeff. (t-value)
aF.F + aK.K
Coeff. (t-value)
MODEL (42)
aF.F + a11
f
gaK.K
Coeff. (t-value)
0.10
(1.8)
0.10
(1.8)
0.03
(0.40)
aF
0.37
(2.9)
0.39
(3.1)
-0.09
(-0.60)
a1
-0.01
(-1.5)
0.01
(1.7)
0.01
(0.50)
aK
0.01
(0.2)
0.06
(0.4)
0.06
(0.94)
a
4.33
(11.7)
4.57
(7.1)
4.28
(12.70)
+0.12
(3.0)
0.10
(2.6)
-2.02
(-0.45)
R2
0.6050
0.6060
0.5354
e'e
4.02655
4.01686
4.73648
Assymptotic Fstatistics
d.f. (n1,n2)
5.29
(5,77)
5.24
(5,77)
(5,77)
Ciritcal-F
3.30
3.30
3.60
df1(n1,n2)
(5,77)
(5,77)
(4,77)
Conclusion
Reject H0
Reject H0
Reject H0
*Assymptotic
8.94
F-statistics computed assuming that the Model (24) wflh
hypothesis to be correct (see Table 9 or Table 12)
124
is the constant term a.
at c = 0.05 and c
The coefficients of P and F are significant
0.01, respectively.
The most inconsistent result
from Models (40) and (41) with respect to the Sadan model is that the
coefficient of (f-g)2 is positive and significant at ci. = 0.01.
Cubic and Quartic Approximations to Sadan Model
The rejection of log-linear process-functions (or the separable
parts of the production function) was indicated as a likely possibility
in explaining the failure of parabolic cylinders as approximations to
the Sadan Model.
This opened up the possibility of using nonlinear
parts as process-functions, and to seek to refute the Sadan hypothesis.
The linear and nonlinear weak separability results were reviewed in
determining what kind of nonlinear process-functions to use in approxi-
mating the Sadan Model.
There appeared to be three possibilities for the model under fixed
K/N ratio assumption with F, I, K inputs.
These three models represent
the three process-function possibilities represented by three iriseparabilities (F,I), (K,I) and (F,K).
The first two inseparabilities
appeared to be successful in the linear restriction tests and the last
two in the nonlinear restriction tests performed previously.
Table 14 indicates the results for a process-function with a nonlinear KI term.
That is, the process-function for the husbandry
process here is assumed to be, g = aKK + a11 + aKIKI
process function here is, f = aFF.
and the biological
The results of Model (43) in Table
14 indicates that all the parameters of the model are significant at
125
Table 14
Approximations of Sadan Model using KI Nonlinear Term
in the Process Function, g
(fg)r,
V = a + f + g +
f = aFF + aPP. g
MODEL (43)
Hypothesis
H0: r= 2
aKK
MODEL (44)
H0: r = 4
(Quartic)
a11 + aIKIK
MODEL (45)
H0: r = 1.5
(8th Order)
(Cubic)
Parameters(tvalues) Parameters(t.values) Parameters(t-values)
Variable
or Terms
a
5.30
(12.7)
5.30
(13.6)
5.30
(12.5)
ap
0.13
(2.6)
0.14
(2.7)
0.13
(2.6)
aF
0.23
(7.3)
0.22
(8.1)
0.23
(7.5)
a1
0.51
(5.0)
0.52
(5.4)
0.51
(5.0)
aK
0.24
(4.4)
0.25
(4.2)
0.24
(4.6)
aKI
0.09
(5.5)
0.09
(5.5)
0.09
(5.1)
-0.21
(-.9)
-0.23
(-.7)
-0.21
(-1.01)
R2
0.6910
0.6946
0.6903
e'e
3.15035
3.11336
3.15710
126
a =
0.01 and positive except for n, the coefficient of (f-g)2, which
is negative though not significant at a
= 0.01
Model (43) is a quartic model since KI is squared in this case.
It should be noted that the inverted parabolic cylinder in Y-f-g space
is indicated by this model; however, it is not confirmed as
is
The parabolic cylinder, however, may not
negative but not significant.
quite have the 'sharp-ridge' property necessary in mimicking the Sadan
Model:
v
mm
(f,g).
Model (44) indicates the sharp-ridge parabolic cylinder:
y
= a + f
g +
r
(f-g)4 [Figure 12 (a)].
The standard errors of the
parameters decrease at the cost of standard error of n.
This result
may be interpreted to indicate that the parabolic cylinder fits the
observations better because it has "flattened" near the ridge and falls
off rapidly with movements away from it.
Because the "flattened"
surfaces do better, the standard error of ri increases indicating in-
creased uncertainty about the surface being "inverted."
Model (45) on the other hand, attempts to approximate a "ridge"
more closely away from the "ridge" and only approximately at the "ridge"
itself [Figure 12 (b)].
127
(a)
y
(b)
Sharp ridge
Optimal
Figure 12:
/
y
Flat ridge
Optimal
a) Eighth Order Approximation
b) Cubic Approximation
All the coefficients in Models (43), (44), (45) are positive and
significant at
0.01 except the parameter r.
It is clear, though,
that for the computed f and g, the observed-computed coordinates (y,f,g)
tended to lie more or less along a straight line, in y-f-g space as
shown below:
--'5o1ic cylinder
ted parabolic
g
al process mix
Figure 13:
Surfaces to Fit Along Scattered Points Around AB
Table 15
Approximation of Sadan Model using KI Term in
the Process Function g
Y = a + f + g +
Hypothesis
Terms
(f_g)r
MODEL (46)
MODEL (47)
MODEL (48)
H0:r=6
H0:r=1.3
H0:r=1.1
Parameters (t-value)
Param.(t-value)
Param.(t'-value)
a
5.25
(13.85)
5.33
(12.43)
5.34
(12.43)
a
0.14
(2.74)
0.13
(2.58)
0.13
(2.58)
aF
0.22
(9.44)
0.23
(7.53)
0.23
(7.56)
a1
0.50
(5.93)
0.51
(4.98)
0.51
(4.96)
aK
0.24
(4.03)
0.24
(4.74)
0.24
(4.90)
aIK
0.09
(5.98)
0.09
(5.05)
0.09
(5.02)
-0.37
(-0.49)
-0.21
(-1.02)
-0.21
(-1.01)
n
ee
0.6993
0.6900
0.6897
3.06519
3.15997
3.16318
129
Table 15 (cont.)
Terms
MODEL (49)
MODEL (50)
r=1.60
r=1.75
Parameters (t-values
Parameters (t-values)
a
5.32
(12.52)
5.32
(12.59)
a
0.13
(2.58)
0.13
(2.59)
aK
0.23
(-1.44)
0.23
(7.38)
a1
0.51
(4.99)
0.51
(15.00)
aK
0.24
(4.57)
0.24
(4.50)
aIK
0.09
(5.06)
0.09
(5.06)
-0.21
(-1.01)
-.21
(-0.98)
R2
0.6905
0.6906
e'e
3.15575
3.15375
130
The results for the other values, of the exponent r of (f_g)'" are
shown in Table 15.
The results indicate that the data is explained
better by a model with r > 2.0 rather than by Model (45), where r
1.5.
However, the coefficient
increased.
gets progressively worse as r is
This indicates and supports the diagramatic representation
of the Sadan thesis in Figure 13.
The other two models with El and FK
terms in the process-function are shown in Table 16.
as well as the previous models from (40) to (50).
They did not do
It may be indicated
that in Model (51) the coefficient of FK is negative and significant
at a
0.05 in spite of being negative.
Furthermore, the parameter, r,
is large, positive and significant at a = 0.01.
This model contradicts
both the production theory and the Sadan thesis, because aF, a1, aK,
aFK are all significantly negative and r is significantly positive.
Similarly, in Model (52) with Fl term in the process-function, the
coefficient of K alone is insignificant at a = 0.01.
The parameter,
is positive and significant in this model at a = 0.01, indicating that
the inverted parabolic cylinder is not inverted at all.
Thus, both
Model (51) and Model (52) fail to support the Sadan thesis; Model (51)
also appears to contradict production theory.
Statistically, the cubic
model with KI term in the process-function, g, seems to have the
highest R2, smallest sum of squared error-term, and almost all the
coefficients a,
aF, a1, aK significant at a = 0.01.
Though i is
negative, it is not significant at ci = 0.05.
We may conclude that the Sadan thesis is not rejected because ri
is not positive and not significant.
If the optimum efforts are
applied, in both processes, we would expect that most of the farmers
131
Table 16
Approximation of Sadan Model using Fl and FK
Terms in the Process Function g
Y
a + f + g + n (f-g)2
g=aFF + aKK + aFKFK
MODEL (51)
Terms
Param.
(t-values)
g=aF + a11 + aFIFI
MODEL (52)
Param.
(t-values)
2.42
(10.40)
4.22
(11.28)
0.12
(2.00)
0.15
(2.80)
aF
-0.10
(-1.69)
0.47
(3.47)
a1
-0.04
(-3.40)
0.25
(2.98)
-0.13
(-4.14)
-0.05
(-0.75)
0.07
(3.06)
a
aKI
-
aFI
aFK
Ti
-0.04
3.03
(-2.78)
(1.85)
-
0.79
(2.55)
R2
0.6328
0.6748
e'e
3.74361
3.31533
132
would operate along the "ridge' as shown in Figure 13.
Thus, as long
as the points (y-aP,f,g) in the process outputs-yield space are
scattered tightly along the line lB (which has a slope of 45° with
respect to both f and g axes) the parameter n can be varied considerably without changing the results too much.
This is indeed what is
suggested by the results of Models (43) to (50).
The fact that the estimated coefficients a,
aF
aK, aKI, a1
are very insensitive to the actual value of the parameter, r, can only
be explained by assuming that the values of the variables, Y-aP,f,g,
lie sufficiently close to lB.
can then be explained.
What is happening in Models (43) to (50)
It is noted that the points lying along a
straight line can be contained in a plane, a cone or a parabolic
cylinder of proper orientation.
In the case of the parabolic cylinder,
one can change from a large positive to a large negative value and thus
invert the cylinder.
Alternatively, one can "flatten" or "sharpen" the
ridge of the cylinder by raising or lowering the value of the parameter
r.
The results suggest that values of the variables Y_aP
f and g lie
along a "ridge" (AB), thus indirectly supporting Sadan's thesis.
133
Neative Error Models and Process Functions
On the basis of previous results, it was concluded that there
exists perfect process complimentarity between the husbandry and the
biological processes.
The biological process was characterized by the
inputs, fertilizer, F and rain, P.
The husbandry process on the other
hand, was characterized by the inputs, capital services K, and the irrigation services, I.
The other methods of combining the various inputs,
such as F and I in the biological process description and K in the
husbandry process, were not as successful in terms of R2s.
Under Sadan complimentarity, it is appropriate to use the negative
error model directly to estimate the process-functions.
To make the
results of the negative error models comparable to the previous models,
the K/N ratio was again assumed constant.
The temperature index I
was again deleted from the regression.
The results from the negative error models for the husbandry process function and the biological process function
Tables 17 and 18.
are indicated in
In Table 17, the husbandry process function has
inputs I, K and also P, whereas the biological process function has
P and F.
The precipitation input is common to both the process-func-
tions primarily because it represents a weather variable, separable
from both the husbandry inputs K,I and biological input, F.
The results in Table 17 indicate that the estimated husbandry
process function has
function has R2
= 04642, whereas the biological process
0.5918.
This indicates that both the fertilizer
input, F, and the capital and irrigation service variables, K and I,
134
Table 17
Negative Error Models with KI Term in Husbandry
Process Function
Husbandry Process
Function (53)
Biological Process
Function (54)
Variables
P
Coeff.
(t-values)
Coeff.
(t-values)
0.09
(1.38)
0.10
(1.87)
0.23
(11.00)
F
-
I
0.88
(5.70)
-
K
0.65
(6.98)
-
KI
0.16
(5.74)
-
C
6.76
(12.80)
4.06
(27.41)
R2
0.4642
0.5918
e'e
5.46261
4.16175
1.31
1.71
d. f.
DurbinWatson
135
Table 18
Ninety-five Percent Confidence Limits on the Coefficients
of Cubic Approximation Model and Negative Error Process
Function Models
Coefficients of
Cubic Approximation Model
Variables
MidValue
5.30
P
Coefficients of
Husbandry Process
Coefficients of
Biological Process
4.88
MidValue
5.72 6.76
0.13
0.08
0.18
0.09
F
0.46
0.40
-
0.52
-
I
1.02
0.82
-
1.22
0.88
0.73
-
1.03
-
-
K
0.48
0.37
-
0.59
0.65
0.55
- 0.75
-
-
KI
0.18
0.14 -
0.22
0.16
0,13
-
-
-
0.00
-
-
-
Constant,
a
-0.42
9(Of
6.23
MidValue
7.29 4.06
3.89
4.21
0.02
0.16
0.10
0.09
0.11
0.23
0.21
- 0.25
C.I.
-0.82
0.6903
0.4642
C.I.
-
-
0.19
C.I.
0.5918
respectively, explain roughly the same amount of variance in yield and
all of the coefficients in the process-functions are positive and significant at a = 0.01.
In comparing these results with the cubic and
higher order approximations to Sadan type complementarity, one must be
careful.
That is, if the Sadan thesis is correct, then the process-
outputs f,g are equal; therefore, the approximation reduces to
y = a + f +
g
n (f-g
a
+ 2g.
Thus, the coefficients computed
using the cubic, quartic and other higher order approximations must be
doubled when they are to be compared with the negative error models.
136
The constant terms computed using the negative error models are 6.76
and 4.06, which compare well with the values of a in Models (43)
through (50).
Similarly, the coefficient of P is 0.10 which compares
well with 0.13 in Models (43) through (50).
Table 18 indicates the
range of coefficient values implied by the cubic approximation model
and those computed using the negative error model or process functions.
The results indicate that the estimated 99% Confidence Interval
(C.I.) overlap each other to some extent for all variables except F.
In the case of I the cubic approximation coefficient is higher than the
direct process function estimate while the case is reversed for K.
For
KI the estimated coefficients are within the C.I. from Models (53) and
(54) at c = 0.01.
Thus, except for F, there appear to be no major
differences in direct and cubic approximation estimates.
In contrast to the process functions of Models (53) and (54),
the negative error models were also attempted using a biological
process function with inputs P, F,
I
process function with inputs P and K.
19.
(and the term Fl) and a husbandry
The results are given in Table
Though the biological process function here explains 67 percent of
the variation in yield, the husbandry process can explain only 25
percent of the variation.
More importantly, the effect of the weather
variable, P, appears to be different in the two process-functions.
The
effect of P is significant in the biological process function, and insignificant in the husbandry process at a = 0.01.
coefficient for F equal to 0.31 and that of I
Model (55) has a
is 0.17, both of which are
137
Table 19
Negative Error Models with Fl Term in
Husbandry Process Function
Husbandry Process
Function (56)
Biological Process
Function (55)
Variables
(t.values)
Coeff.
P
0.13
(2.61)
F
0.31
(11.50)
I
0.17
(3.94)
(t-values)
Coeff.
-0.003
(-0.04)
(5.17)
K
Fl
0.04
(4.25)
C
4.26
(29.90)
-
5.59
(11.61)
0.6683
0.2513
ee
3.38191
7.63
d.f.
5,82
5,85
Ourbin
Watson
significant at c
1.73
= 0.01.
1.26
The coefficient of the Fl term is only 0.05,
although significant at ci. = 0.01.
Thus, as far as negative error models are concerned, the processfunctions appear to support the Sadan complementarity hypothesis better
than do the process-functions (Table 19).
Under Sadan complementarity,
both biological and husbandry processes must be optimized so as to
result in equal contribution to yield.
This is indicated by the rather
138
close R2's in case of the process-functions (53) and (54).
In the case
of (55), (56) process functions, the R2's are diverse and the coefficient on precipitation, P, is not the same in these two functions.
Thus again, negative error model estimates indicate that the husbandry
process with inputs K,
I and term KI are more consistnet with the
notion of complementary processes than those with merely K.
Additive Error Model
A cubic or a higher approximation to Sadan model (with perfectly
complementary process functions) is a multiplicative error model:
+nf_gjrc
v = e'= e a+ f+ g
where,
r' N (O,c2), f = aFlnF
g = a1lnI + aKlnK
etc.
The additive
error model on the other hand is:
Y=e my =e
a
+ f + g
flfgr
N(O,a2).
The nature of the true model decides which of these models is
better.
In the case of Cobb-Douglas true model, use of log-linear form
with the multiplicative error terms may be expected to do better.
This
is because the error terms are small in magnitude and distributed very
closely to the normal with zero mean.
On the other hand, if the true
model is not Cobb-Douglas then this superiority of multiplicative error
model may not always hold.
It is possible that the true model is the nonlinear logarithmic
model such as TRANSLOG; if so, the additive error terms model does
better than the multiplicative error term model.
The result of the additive error model is shown in Table 20.
It
indicates that the coefficients are rather stable between additive and
139
the multiplicative error term models.
The coefficients are all
significant at ci. = 0.01, except for the parameter, n.
This parameter
is negative, though with a very large standard deviation, making its
sign to be insignificant at c#. = 0.01.
adjusted
It should be noted that the un-
has increased slightly, indicating that the additive error
model does slightly better than the multiplicative error model.
Though
this result is unlike the results obtained when using a log-linear CobbDouglas model, the difference could very well arise from the fact that
the model is log-nonlinear cubic.
The log-nonlinear model can be
better when the model has additive error terms, probably because the
additive errors are closer to a normal with zero mean.
One important distinction is that the additive error model has a
substantially less negative value for parameter n and the standard error
of
is increased.
This means that there is less certainty about the
downward inverte& nature of the parabolic cylindrical surface in the
y-f-g space as indicated before.
This, however, is as required.
If the
process-mix is optimal, the observations must lie along the "ridge
already discussed.
as
So the data should be indifferent between planes,
cylindrical surfaces and conic surfaces as long as they contain the
uridgeu line (say line AB in Figure 13).
This indeed appears to be
true for the additive error model, more so than for the multiplicative
error models.
140
Table 20
Additive Error Model for Cubic Approximation
to Sadan Type Process Complimentarity
(57) Y = e
Terms
=
ea + f + g +nf-g
1.5
+ E
,
Estimated Coefficient
N (0,a2)
t-Statistics
a
5.29
(14.66)
ab
0.09
(2.39)
af
0.23
(4.15)
a1
0.54
(3.79)
ak
0.22
(3.56)
aki
0.10
(3.89)
-0.09
(-0.32)
0.7002
e'e
3650.12
d.f.
88-7
Durbin-Watson
1.66
Stati stics
141
Model-B Results
The maintained hypotheses of the previous model was that the
weather variables P and T were separable from other input variables,
such as F,
I,
K, N, etc.
Furthermore it was assumed that the manage-
ment process was perfectly complimentary to the farming process, constituting of both the biological and the husbandry process.
The former
assumption relates to the independence of the precipitation and temperature index variables from other farming inputs.
The latter relates
to the behavioral assumption of optimization in addition to separability, and so is distinct from the previous one.
Though weather variables are not possible to be controlled at
present, weather forecasting has advanced consicerably.
This means
that many decisions in farming are done better now, in terms of timing
of inputs.
This of course does not necessarily imply that the inputs
and weather variables are interrelated in the sense the other control-
lable inputs are interrelated.
But because of weather variables being
potentially capable of affecting input decisions, it was felt advantageous to relax this assumption of separability between P,T and F,I,N,K
group.
ference.
The fact that management inputs are not known, does make a difThe previous assumption that the management process be con-
sidered perfectly complimentary to the 'other' process does also appear
realistic and necessary as before.
Though weather variables are fore-
castable, the accuracy of forecasts are not high enough to influence
day-to-day process-control
.
So the management process may be assumed
independent of weather variables.
142
Thus Model-B was estimated maintaining the assumption that the
management process was a perfect compliment to the 'other' process,
where
'other1
stands for the whole gamut of weather, biological and
the husbandry inputs and the processes represented by them.
The pur-
pose of the model is to test if the weather variables are indeed
separable from F,I,N,K the biological and husbandry process inputs.
The separability was tested using only linear restrictions.
In
the presence of linearly restricted separability, it was considered
unnecessary to test for non linear restrictions, because linear restrictions suffice one to confirm the Model-A assumptions.
It must
however be noted that for computational ease, PK, PN, TK, TN terms were
deleted from the regression maintaining the hypothesis of separability
between weather inputs P and T with capital service K and labor service
N.
This appears to be a reasonable assumption, because the magnitudes
of K and N are not usually influenced by weather variables.
irrigation service I could be influenced.
Though
And because terms EN, FK
were not found significant in the previous linear restricted models,
it was deleted maintaining the hypotehsis of separability between F
and N, K also.
The results are indicated in Table 20.
Only the coefficients of
P and I are significant at o= 0.01 and both are positive.
Model
The
(58) is the general translog function with the maintained hypo-
thesis that F, P, T are separable from K and N.
Taking this as the
general model, we can put further linear restrictions so that we delete
PT, PF, P1, TF, TI, El, FK, EN, P2 and T2.
This means Model (58) is
transformed due to linear restrictions into Model
(9): [F,(K,I,N)].
143
The validity of this linear restriction implies that P,T are separable
from [F,(K,I,N)].
The F-statistics of the restriction discussed above was computed
at 2.20.
And because the critical F-statistics at o= 0.01 is 2.80
with (8,66) degrees of freedom, we can safely conclude that the assumption of linearly restricted weak-separability of (P,T) with [F(K,I,N)]
is valid and not rejected.
A further linear restriction could be em-
ployed on Model (9): [F(K,I,N)] by deleting the cross-product terms,
MI, KM. so that the new model becomes Model (19): [F,N,(K,I)J with
double partitions.
The F-statistics of this final model, with respect
to the general Model (58) is computed at 2.00.
F-statistics at
o
Because the critical
0.01 is 2.60 with (10,66) degrees of freedom, we
cannot reject the null hypothesis of separability of P,T,F from
[(K,1),NJ.
Thus, in conclusion, we can confirm that the (P,T) are weakly
separable from [F,N,(K,I)]as was assumed in models with KI terms.
This
result can also be extended to separability between (P,T) and (F,t,M,K),
in general, by expanding Model (58) to include cross-products terms
FM,FK, etc. and re-computing the model.
The new F-statistics computed
is however expected to confirm this, so it was not actually computed.
In any case, because the relevant model is Model (19): [F,N,(K,I)J
amongst Model-A results, we see that (P,T) is separable from [F,N,(K,IJ
and so the results of Model-A remain unchanged, even after losening
the constraint of weak separability between (P,T) and [F,N,(K,T)].
144
Table 21
Separability of Weather Variables P,T and inputs F,I
or Separability Test of [P,T,(F,(K,I,N))]
MODEL (58)
[P,T, (F, (K,I,N)]
Variables
Coeff.
MODEL (9)
MODEL (19)
[P,T, (F,K,I,N)] [P,T, (F,N, (K,I)fl
(t-value) Coeff. (t-value) Coeff.
(t-value)
P
1.630
(2.77)
0.170
(2.85)
0.173
(2.93)
T
-0.180
(-0.58)
-0.009
(-0.27)
-0.007
(0.22)
F
-0.058
(-0.16)
0.121
(0.86)
0.132
(0.99)
I
0.878
(2.28)
0.836
(3.79)
0.682
(3.57)
N
-0.882
(-0.81)
-0.435
(-0.39)
-0.533
(-1.22)
K
2.816
(1.00)
0.573
(0.19)
1.595
(0.67)
p2
-0.138
(-1.28)
-
-
T2
0.089
(2.42)
-
-
F2
-0.009
(-0.52)
-0.006
(-0.35)
-0.006
(-0.41)
i2
0.001
(0.17)
-0.003
(-0.87)
-0.003
(-0.95)
N2
0.012
(0.09)
-0.008
(-0.06)
0.053
(1.20)
K2
0.295
(0.70)
(0.04)
0.113
(0.49)
PT
-0.169
(-1.63)
-
-
PF
-0.031
(-0.44)
-
-
P1
0.012
(0.27)
-
-
TF
0.054
(0.94)
-
-
TI
-0.065
(-1.49)
-
-
Fl
0.023
(1.39)
-
-
NI
-0.026
KI
0.111
NK
C
(-0.80
0.0016
-0.041
(-1.33)
(1.34)
-0.195
(3.93)
-0.159
(-.39)
-0.040
(-0.10)
8.174
(1.49)
5.281
(0.95)
0.128
(3.37)
-
7.550
(1.48)
R2
0.7709
0.7095
0.7016
e'e
2.33566
2.96114
3.04214
F-statistics (df)
D.f.
None
88-22
2.20 (8,66)
88-14
2.00(10,66)
88-12
145
VI.
SUMMARY OF RESULTS
The results of the Chow test indicated that eastern and western
Oregon indeed have two distinct wheat production functions.
productivities are also different in these two regions.
The factor
Water pro-
ductivity appears to be small or near zero in western Oregon.
Ferti-
lizer and precipitation both appear to influence the yield in a positive manner in both regions.
The coefficients also appear to be com-
parable to one another for these two variables when we compare Model
(6) with Models (43) through (50).
This is remarkable considering the
fact that the types of wheat grown in eastern and western Oregon differ
significantly from one another.
The coefficients for capital service,
K, also are comparable in value between Model (6) and Models (43)
through (50).
Model (6) however, suffers extensively from multi-
collinearity between K and N; consequently, the results may change
somewhat if multicollinearity is reduced when K/N is fixed as before.
The test of linear homogeneity of the production function failed
for eastern Oregon.
This was, of course, anticipated, since we did not
expect that the "per unit land" model itself could be linearly intensified.
Linear homogeneity of the per unit production function would
imply that there is increasing or decreasing returns to scale.
The
exact value of returns to scale is of some interest, although the local
nature of the result reduces its value considerably.
Thus returns to
scale here would not be difficult to interpret, as it would refer to
146
scale implying process intensities.
But the computation of returns
to scale or intensities has not been attempted presently.
The conclusion derived from the previous results chapter is that
the linear restriction of weak separability is valid between (P,T) and
(F,I,N,K) in eastern Oregon.
Thus the maintained hypothesis of
Model-A is a valid hypothesis.
Within the Model-A results, the test
of single, double and pairwise partition using linear restrictions Tndicates that not all F,I,N,K are strongly separable.
The linear re-
strictions where Fl and KI terms were absent were usually rejected under
the appropriate F-test.
Thus the ambiguity arose as to the empirical
superiority between log-quadratic models with Fl and KI terms.
The
model with KI term has a slight superiority in terms of the sum of the
squared errors compared to one with the Fl term, as indicated by the
results of the Models (14), (9), (19), (27) versus those of Models (13),
(10), (16), (25), respectively.
But statistically speaking the F-test was still ambiguous in the
sense that both these models could not be rejected on statistical
grounds at ci = 0.01.
The ambiguity persists in all the log-quadratic
models with linear weak separability restrictions.
The nonlinear weak-separability restriction tests using assymptotic
F-distributed statistics again gave similarly ambiguous results as
regards Model (39) where K and I were not nonlinearly separable and
Model (38) where F and K were not nonlinearly separable, as shown in
Table 12.
Thus, the best process-functions could not be decided upon,
as ambiguity presented in both the linear and nonlinearly restricted
models.
This led to the further restriction of Sadan perfect process-
147
complimentarity, to be applied to the log-quadratic, TRANSLOG.
However
as indicated by the models (40), (41) and (42) of Table 13, the hypothesis of perfect-process complimentarity using the Cobb-Douglas approximation to the process functions were rejected quite strongly, in all
cases.
As already explained this rejection implies that either the CobbDouglas process-functions were invalid or the process complimentarity
was low.
The previous rejections of pairwise strong separability,
however, seemed to indicate that the assumption of Cobb-Douglas
process-functions was responsible for this rejection.
Thus other log-
quadratic process-functions were necessary to explain the data better.
.Three models with log-nonlinear process functions were then attempted
using cross-products (Fl), (KI), (FK) in the process-functions.
Models
(51) and (52) for FK and Fl, respectively, in Table 16 did rather poorly in approximating the Sadan model, which is also inconsistent with
general production theory.
On the other hand, Models (43) through
(50) in Tables 14 and 15 strongly support the Sadan thesis of perfect
complementarity, where the husbandry process consists of a simple log-.
quadratic in K and I and the biological process is a simple Cobb-Douglas
in P and F.
The coefficients of Models (43) through (50) are all positive and
significant except for the parameter n which was negative and not
significant at c = 0.01.
This indicated that the observed data lie
along a straight line 'tridge
Y = Mm
(f,g).
characteristic of the Sadan function,
Thus the higher-order functions of Models (43) through
(50) have enough flexibility to depict a situation which represents
possibly a situation of near perfect process complementarity.
The
difficulty of attempting to resolve the close competition between
models with Fl and KI terms can best be understood with the help of
Graph 2 between Fl and KI given below.
As can be seen, the log-quad-
ratic forms with Fl would do nearly as well as that with the KI term,
because the two terms are very strongly correlated.
But the cubic and
higher-order functions used to approximate the perfect process complementarity also have highest R2 or least sum of the squared errors,
when the process-functions have KI term.
The validity of Sadan's perfect process-complementarity allowed
the use of direct estimation of the process-functions using the framework of negative error-models.
Again, the results from the estimated
process-functions are consistent with the cubic approximation model in
general.
The implied true coefficients are obtained from the cubic
approximation model by doubling them (except for P and the constant)
so that they are comparable to the process-function coefficient estimates.
Except for the coefficient of F, the results can be considered
to be mutually supportive of each other.
The cubic approximation model
does, however, have a coefficient of F twice as large as the coefficient in the direct estimation of the process-function.
Graph 2:
'(I
KI vs. Fl, Eastern Oregon
67.7'2
0
f,3. 795 0
I
I
e
5S.2I '
I
I
5t.3'. '
V
V
V
43. I59 I
V
0
V
V
S
S
I
27.910
p
23.923 5
S
p
$
0
I
.2'
V
15.91,9 P
P
e
I
PIP
11.9F'2
7.97'.
.3 P.
5
'2
3.901
2
.3
32
-.000 .X
.
0.900
j0.fl'.
?fl.650
3I.??
52.440
'.t.0(.1
41.143
150
Total and Marginal Productivities based on the Cubic Approximation to
Sadan's Perfect Process Complimentary Model
The cubic approximation in Model (45) was used to compute the
various input-output relations conditional upon other factor inputs
being fixed.
For Figure 14, precipitation was assumed to be fixed at
10 inches and capital at $10.00/acre.
Irrigation service level
I was
given one of the three values 1.0, 0.5 or 0.07 and the yield computed
for varying levels of fertilizer inputs.
Figure 14 shows that the
marginal physical product changes continuously for fertilizer from
more than 5 bushels/lb. to 0.25 bushels/lb.
diminishing marginal product.
Fertilizer input has a
The figure indicates various yield
curves at differing levels of irrigation service.
The level
is taken merely to indicate the sample mean situation.
I = 0.07
The curves
are unrealistically reduced to the F-axis if I = 0.0.
To indicate the effect of increasing capital service on yield,
another set of fertilizer yield response curves were computed using
Model (45) as shown in Figure 15.
The important fact here is that
at lower levels of irrigation, say at I = 0.07, the fertilizer responsefunction does not change at all from the previous curve in Figure 14.
This is due to the fact that at low or no irrigation, increasing the
capital service alone does not improve yield.
The husbandry process
of seed bed preparation, tillage and harvesting does not improve
yield without increasing the irrigation service.
be obtained for I at low levels of K.
Similar results would
However, when the irrigation is
151
Fertilizer Input, Marginal Product and Expected Wheat Yield
Based on Cubic Approximation
Model (45): P = 10 inches, K = $10.00/Acre.
Yield (10 Bushels)
----Marginal Product (Bushels per lb.)
5
4
3
2
I
0
50
F
100
150
Fertilizer Input (lb./Acre)
Figure 14
200
250
152
Fertilizer Input, Marginal Product and Expected Wheat Yield
Based on Cubic Approximation
Model (45):
P = 10 inches, K = $20.00/Acre
Yield (10 Bushels) --- Marginal Product (Bu./lb.)
F
Fertilizer Input (lb./Acre)
Figure 15
153
present,
I
= 1.0, the effect on yield is substantial.
For comparison,
in the usual range of fertilizer application in eastern Oregon,
say 50-100 lbs/acre, the marginal product of fertilizer is in the
The corresponding
range of 1.75-1.10 bu./lb. when K = $20.00 per acre.
values when K = $10.00/acre were 1.3-0.75 bu./lb.
The curves of
Figure 14 and Figure 15 cannot be directly used to compare marginal
productivities of irrigation service I.
At K = $10.00, the total
gain in yield from irrigation appears to be in excess of 10 bu./
acre beyond 40 lb./acre fertilization.
Similarly at K = $20.00, the
total gain in yield from irrigation appears to be in excess of 15 bu/
acre beyond 40 lb.Jacre.
At higher level of fertilization, F = 120
lb/acre, the total gains in yield are 15 and 20 bu./acre, respectively
at K = $10.00 and K
$20.00.
The total gains in yield are relevant
here because the decisions relate either to have irrigation or not
to have irrigation.
However, the cubic approximations are local ap-
proximations and therefore the results are to be interpreted accordingly, with care taken not to extend the results too far beyond the
locality of the approximation.
The case in point is the value of
I = 0.0, though this value is legitimate ordinarily, the log-transformation of this value tends to be negative infinity; and so the lowest
value for the curves has been taken as I = 0.07, the sample midpoint
for irrigation.
This value should correspond more or less to the sit-
uation of dryland wheat production in eastern Oregon.
The negative error models could also be used.
However, the total
yield estimated using such a model will be definitely biased.
The
marginal products for the negative error models therefore cannot be
154
computed without knowing the unbiased yield, since the marginal
product terms contain yield in their expression.
of substitution
The elasticities
could also be determined for the inputs using the
process functions; however, later on they will be shown to be unnecessary for the present case.
It may be noted here that direct estima-
tion of process-functions does not fulfill the purpose of measurinq
marginal productivities unless there are means by which the yield
can be estimated without bias.
Such an unbiased estimate of yield
has been obtained by using the cubic approximation here.
Though the previous figures indicate that there are diminishing
returns for all inputs F, K, I, the substitutions are difficult to
stipulate using those "conditional" curves.
To simplify matters, the
precipitation and irrigation levels were fixed at P
I
= 1.0.
10 inches,
Then using a price for fertilizer of about $400.00 per ton,
the expenditure on the fertilizer and the capital service was fixed
at $10.00/acre and $20.00/acre.
It should be noted that the cost of
irrigation, etc. are ignored at this point for simplicity, because
their explicit consideration does not change the conclusions drawn.
Figure 17 indicates the result of changing the K/F ratio on total yield
(keeping the expenditure constant).
It may be noted that for the maxi-
mum yield the K/F ratio is around 0.6.
The maximum yield that was
attainable at $20.OoJacre expenditure was 37.60 bu./acre and the maximum yield that can be achieved at $10.00/acre was 26.2 bu./acre.
Since, keeping I = 1.0 implies that, K can represent the husbandry
process-function (or
its monotonic transformation in this case, g)
and F represents the biological process-function (or its transformation)
155
Cross-section of the
Yield
Expenses on F and K
Assumption:
I
Cubic Approximation Function
vs. K/F Ratio
Fixed at $10.00 and $20.00
1.0 and P = 10.0 inches.
Yield
(Bushels/Acre)
45
40
35
30
25
20
15
0.05
0.1
0.2
or
.04
0.6
Process Mix Ratios
Figure 16
1.0
2.0
4.0
6.0 8.0 10
156
f, the curves in Figure 17 can also be regarded as the cross-sectional
profile of the inverted parabolic cylinder in the Y-f-g-space.
One
peculiarity of this form of the cubic approximation is that the between
process-substitution is highly elastic near optimal process-mix and
highly inelastic away from the optimal process-mix.
This can be ob-
served from the yield - K/F ratio curves in Figure 17.
The maximum
yield attained changes rather slightly between 0.4-0.9 of K/F ratio,
but begins to fall off rapidly beyond these ratios.
The closer in-
spection of the cubic approximation function, Y = a + f + y +
For the above functional form it can be shown
reveals the reason.
that the process-substitution between f and g,
fg'
takes on the
following simple form:
YfYg (Yf.f + Ygg)
(Yfg=
f.g (Yf_Yg)LYfg
where, (YfYg)2
fa
= 1-r2
=
r2
2(fg)21),
Yfg = r(r-1)n
(f-g)
r- 2
,
and
2(fg)2(r-1)
The denominator term tends towards zero when f and g are close together, so that near optimal process-mix,
fg
tends to infinity.
On
the other hand, as f and g diverge from one another, the YfYg term
becomes small and
fg
tends towards zero.
Within Process-Substitutions
Within the framework of Sadanrs perfect process complimentarity
and the assumption of optimizing behavior, we can write down the process-functions as follows:
157
(1)
Biological Process Function
= q
b
+ hF, and
(ii) Husbandry Process Function,
Y =
where,
cients.
7,
,
F, T,
a + bP + ci + dK + yKI,
are all
in logarithms and a,b,c ..... are coeffi-
The within process substituability between inputs in a Cobb-
Douglas production function is known to be unity.
Thus for the bio-
logical process function, since there is only one man made input F,
unitary substitution elasticity with respect to P does not have any
practical significance (since P is not a controllable input).
On the
other hand in the husbandry process function, the substitutability
between K and I is of some interest.
The husbandry process-function, though a simple log-quadratic,
does have unit elasticity of substitution between K and I just as
in the Cobb-Douglas function form.
This may be shown by computing
the elasticity of substitution between K and I,
aKI.
We note that,
GKGI (GKK+GII)
KI=
KI (GI2GKK_2GIGKGIK+GK2GII)
where, Y = G(P,I,K) represents the process function in original variables, denoted without a bar on top of the respective log-form variables.
and GKI represent the derivatives of G with respect to K
and with respect to K and I, respectively.
tives, G1, G11, and
GK
Simiarly for other deriva-
etc. it may be noted that,
(d+glnI)(Y/K), G1 = (c+glnK)(Y/I)
GKK = (d+glnI)(d+glnl-1)(Y/K2) = (GK2/Y)
(GK/K)
158
G11 = (c+glnK)(c+glnK-1)(Y/12) = (G12/Y)-(G1/I)
GIK = (c+glnK)(d+glnl)(V/KI) = (GIGK/Y)
So that, the numerator _GKGI(GKK+GII) in the expression for
is as follows:
_GKGI [(d+glnI)(Y/K) K + (c+glnK)(Y/I)I]
=
_GKGK [(c+d) + g(lnK + ml)] Y
The denominator KI [GI2GKK2GIGKGIK+GK2GII] in the expression for
takes the following form after simplification:
(IK)(c+glnK)(d+glnI)[(c+d)+g(lnK+lnI)](Y3/I2K2)
Thus the numerator and denominator are equal, implying c<1 = 1.0 within
the husbandry process.
Similarly it can be shown that the elasticity of substitution
between P, and I and that between P and K is unity.
Thus, within
process substitution elasticities in this Sadan model are just like
those in Cobb-Douglas production function.
159
CflNCI IIT(ThI
The conclusion of the present study can now be reiterated as
follows:
a) There exists weak separability amongst the biological, weather
and husbandry inputs in both eastern and western Oregon wheat production functions.
The implication is thus clear that in aggregating
into higher level of aggregate production functions, the preferred
approach would be aggregation along these implied processes rather
than across farms.
b) There also exists empirical support to the notion that there
are some processes in agriculture which are
nearly perfect compliments
to one another.
c) There are major differences in production functions in eastern
and western Oregon, resulting from the differences in precipitation
and other climatic factors.
For the eastern Oregon data, set (K,I) has been found inseparable,
in the sense that the hypothesis of separability of K and I is consistently rejected in both linear and nonlinear restrictions tests.
Thus
K and I belong to a single process within which they interact condu-
cively to increase yield.
one another but not with F.
They can therefore be aggregated with
The effects of K and I on yield can be
summarized with a single index g constructed from K and I and incorporated into an aggregate production function, Y =
q(f(F),g(K,I).
This kind of aggregation would not result in any aggregation bias.
In other words, if the aggregation is performed using K and F where
160
they constitute an aggregate "expenditure" there is a good possibility
of aggregation bias.
On the other hand, in the case of the western
Oregon data there appears to be more than one good way to aggregate
inputs to make an aggregate index.
Thus additive separability of western Oregon Cobb-Douglas function indicates that an aggregate 'expenditure' constructed from K
and F is valid, and results in no aggregation bias in estimating the
c-function.
There are of course other equally valid aggregation pos-
sible, such as aggregation of K and I, and of I and F.
Also, it may be noted that in eastern Oregon the husbandry process with function, g(K,I), and the biological process with function
f(F), are highly complementary to one another, as Sadan has hypothesized.
The process-functions when estimated directly using the nega-
tive error model gave rise to similar coefficients for the variables
as was obtained in the cubic approximation model.
However, though this
provides some support to Sadan's perfect process compi mentarity, the
biased nature of the intercept terms disqualifies these models for
estimating marginal productivities directly, unless the yields are
known without bias a priori.
Thus the conclusion that can be drawn is that Sadari's thesis of
complimentary processes appear plausible and is at least not rejected.
The important consideration here is that the cubic approximation was
necessary to attempt to refute Sadan's thesis..
strictly local.
The results are
When refutation of a thesis is sought, this can be
attempted locally since local refutation is a valid refutation.
161
On the other hand, the refutation may not occur outside of the
locality.
Thus Sadan's model must not be regarded as a global result
for eastern Oregon.
The same conclusion also holds for the weak
separability tests for eastern and western Oregon wheat production
functions.
162
LIMITATIONS AND RECOMMENDATION
The analysis in this present study has maintained a few hypotheses and the result of the analysis are conditionally valid upon them.
The important assumption has been that the "management process" is
perfectly complimentary to "other processes" and that farmers behave
optimally in equating these process-outputs.
This is Sadan's thesis
with respect to intangible inputs such as "management".
If Sadan's
process-complimentarity does not hold here, then there certainly will
exist some specification error in the present estimation of the production function.
Since, test of Sadan's thesis in this context must
await the solution to the measurement of management variables, the
present study results must be viewed with some skepticism.
fication error may be quite large.
The speci-
This is the major methodological
limitation of the present study.
Related to this Sadan type process complementarity assumption is
the assumption of linear homogeneity or physical replicability of the
wheat production in space.
When the "management" inputs are the
bottleneck to expansion and/or when technological change allows
conserving management inputs, the assumption of linear-homogeneity
may not hold.
This is particularly true when the 'management" inputs
are not available for 'hire".
Thus, Sadan type complementarity which
implies weak separability as well as perfect process complementarity,
may not hold.
Linear-homogeneity may be physically possible, but not
so mangerially; this may result in specification error.
163
The tests of separability and linear homogeneity and processcompernentarity are all based on the quadratic and cubic approximations to a general functional form at a point.
The results are there-
fore only true locally and may not hold far beyond the point of approximation.
This is a serious limitation of the present methodology.
In the absence of a priori knowledge of the form of the production
function, this limitation appears very difficult to overcome.
Another equally important limitation of the present study stems
from the nature of the county level data.
Since the data is already
aggregated amongst farms in the county, the relevant question that can
be asked of this data related to aggregation beyond the county level
data.
It is felt, however, that when the data is available at lower
levels of aggregation, the present methodology can be re-employed to
see if the presently used aggregation to the county level data has
results in significant aggregation bias.
The analysis would have considerably benefited if disaggregate
county level data could have been obtained for totally, partially and
not irrigated farms.
This would have permitted greater accuracy in
the measurement of irrigation service, I.
The measurement error of
capital and labor services utilized in this study is quite high,
since total fuel and oil expenses and hired labor wages were used.
Disaggregated data specific to wheat production activities would be
advantageous.
The problem of multicollinearity between K and N. how-
ever, may still persist requiring the use of the fixed K/N ratio assumption or ridge-regression techniques in the analysis.
Similarly a
164
better temperature index is needed which relates to the minimum
temperature in eastern Oregon.
In view of the above discussion the following recommendations
are appropriate:
a) The analysis of production relations at a given level of aggregation can and should be conducted to test for weak and strong separabilities.
This opens up the possibility of identifying or ascertaining
various salient features of Droduction technology, such as its
analysis into distinct and separate processes and sub processes.
The
identification of such separabilities or processes comprises the first
step towards valid aggregation at higher level of analysis;
b) The present analysis can be extended to other crops and activities of the farm.
When var.ious processes are identified within each
activity, the appropriate aggregate index for each can be formulated
so that aggregate production functions for cropping, dairy and other
major activities can be estimated without too much aggregation bias;
c) The use of a cubic approximation to the Sadan model has mdicated that the nonlinear-in-parameter forms are economical in parameters.
Thus beyond one or two variables cases, this economy in para-
meter becomes advantageous from the computational standpoint in providing necessary functional flexibility.
The perfect process-comple-
mentarities are of some interest from the standpoint of simplifying
production function estimation procedures.
The use of higher order
approximations beyond quadratic can be used in the analysis of high
process complementarities;
165
d) The use of separability tests on second order approximations
to a general production function at a point, can lead to ambiguities
regarding the process-functions.
The option is either to extend the
analysis to higher order-approximations or to proceed with notions such
as "perfect complements" or "perfect substitutes" amongst processes.
The use of the latter is to provide the simplest means of upgrading
from second order to third, or higher order approximations that may
become desirable, without exhausting all of the higher order approximation possibilities.
This approach leads to choosing a functional
form dictated by the economic characteristics of the production process
and not vice versa.
166
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1961, 43, 876-88.
Moore, F.T.
Economies of Scale:
Some Statistical Evidence.
Journal of Economics, 1959, 73, 232-245.
Quarterly
Mundalak, Y.
Empirical Production Functions Free of Management Bias.
Journal of Farm Economics, 1961, 43,44-56.
Nataf, A.
Sur La Possibilite de Construction de certains Macromodeles.
Econometrica, 1950, 16, 232-244.
Pollack, R.A.
Conditional Demand Functions and Consumption Theory.
Quarterly Journal of Economics, 1969, 60-78.
Index Numbers, Annual Average, U. S.,
Price Paid by Farmers:
Extension Economic Information Office, Oregon State
1950-1977.
University, Corvallis, Oregon, 1979.
Conditional Demand Functions and the Implications of
Separable Utility.
Southern Economic Journal, 1971, 37, 423-433.
Russell, R.R. Functional Separability and Partial Elasticities of Substitution.
Review of Economic Studies, 1975, 42, 79-86.
Ruttan, V.W.
The Economic Demand for Irrigated Acreage; New Methodology and Some Preliminary Projections, 1954-1980, John-Hopkins Press,
1
Sadan, E.
Partial Production Functions and the Analysis of Farm-Firm
Costs and Efficiency.
American Journal of Agricultural Economics,
1970, 62-70.
171
Samuelson, PA.
The Foundations of Economic Analysis, Harvard University Press, 1947.
Schmidt, P.
On the Statistical Estimation of Parametric Frontier Production Functions.
Review of Economics and Statistics, 1976, 58,
238-23 9.
Shephard, R.W. Cost and Production Functions, Princeton University
Press, 1953.
Theory of Cost and Production Functions, Princeton University Press, 1970.
Smith, V.L.
Investment and Production, Cambridge, 1961.
Stegman, E.C.
On Farm Irrigation Scheduling Evaluations in Southeastern
North Dakota, Number 76, Agricultural Experimental Station. North
Dakota State University, Fargo, 1980.
Strotz, R.H. The Utility Tree - a Correction and Further Appraisal.
Econometrica, 1959, 27, 482-488.
Theil
,
H.
Linear Aggregation of Economic Relations, Amsterdam, 1954.
Principles of Econometrics, John Wiley and Sons, 1971.
Theil, H. and A.S. Goldberger. On Pure and Mixed Statistical Estimation in Economics.
International Economic Review, 1961, 2, 6578.
Thomas, H. R.
Economic Productivity of Water and Related Inputs in
the Agriculture of Southern Idaho. Unpublished Ph.D. Thesis.
Oregon State University, Corvallis, Oregon, 1974.
Walters, A. A.
Production and Cost Functions:
Econometrica, 1963, 31, 1-66.
An Econometric Survey.
APPENDICES
172
APPENDIX A
Theorem 1:
If a general function y = F(X1 ..... ,XN) is weakly separable, such
that,
y = F(Xl,...,XN) =
,...,X
{L(XlI,...,XN),...,Lr(X
)}
r
and if F(.) can be expressed as G(x1,... ,xN), where x1 is a monotonic
_cX1
transformation of X1,
for all
1
i.e. x1
x1(X1) and x1
> 0
or
--- < 0,
= l,...,Ni,...,NrN, then, y=G(x1 ..... ,x) is weakly separa-
ble, such that,
r
y = G(xl,...,N) =
Proof:
Weak separability of F(X1,... ,XN) has the necessary and sufficient
condition (due to Leontief) as follows:
'
'i
A)
where,
X' (i.e. X's in rth separable group) and Xk
where r
(B)
0
Xk 'F/X,j
The equation (A) can be further simplified as,
s.
FjFjk
0, where F =d
F:lk
If we represent,
etc.
Fik
--- = x1 for all 1, then, differentiating G with
1
respect to X1 's we obtain,
F
X
=
F
=
Fik = Gik XjXk. Fjk
GjkXjXk
and, from (B),
XXX
(C)
FiFik
Since, x1
0 for all the values of l's, we conclude that,
FiFik = (GiGik
GiGik)
=
173
(D)
GiGik =
GiGik
This is the necessary and suggificient condition for G(x1,.
.
.
,x)
to be weakly separable, so we can write,
I
r( r
{f (k1,... ,XN),... ,f \X,..
I
y = G(xl,...,xN) =
,
r
Q.E.D.
Theorem 2:
If y = G (x1,..
X1 and G(.)
.
,x) is a second order polynomial (quadratic) in
ff (xl,...,XN ),...,f(X,...,Xr )that is, G() is
=
1
weakly separable, then, either
PART-I
r
is linear in fr(X
nr
where f'(..) are second order poly-
nomial (quadratic) in xs, or
PART- II
is 2nd order polynomial (quadratic) in f()'I
where
fr(.)
are
linear in X''s.
Note:
Proof is provided for two separable groups here, x
,XK
and Q = {xk+1,.
.
. ,X,.. .X
The proof can be extended
.
into three or more partitions by induction.
Consider indices
i,j = 1,. ..,k;p,q = k+1,...,N and m,l = 1,2,... ,K,. . .N.
Since G(x1,.. .,xN) is quadratic we can express G() as,
+ X' X
y =
N
(E)
N
'1
+
E
rn
where,
=
and
(
n"1N
N1" kN
and y is symmetric square matrix.
x1,... ,x,
1m
174
The Leontief condition for Weak Separability is,
(F)
=
GiGik
Note that, for a quadratic G(), we have,
N
= (c*+
Gik-
1'y1X1 '1,
G,j
=
Z
(
)
ik
So that (F) implies that,
N
(G)
1
j
'k
ik
(1.iiY.
jk
)+
j kik
l
=
The necessary and sufficient condition for (G) to hold is, either,
the LINEAR RESTRICTIONS:
(H)
ik
''ik
= 0 , for all
Si,j=1,..,k
k=k+1
, . . . ,''
or, the Non-Linear Restrictions:
'i,j = 1,... ,k
:-Y.
--
(I)
=
_a
and
=
forp,q=k+1,.. .,N
1qm
(1,m =
Linear Restrictions give rise to the validation of PART-I
Proof:
PART-I
Rewirte (E) as follows:
K
N
+
y =c
+
c
k+1
X+
pp
KN
+E
i
KK
.y.. X.X.
.
3
N
N
k+1
k1
X.X+
1
k+1
p
Note that Linear Restrictions implies that
and p.
x=
pq
=
0 for all
i
So we can rewrite the expression in terms of
(X1,. . .Xk),
y
1
+OL'X
k'
+ X' X +'&
ij' { =
AAA
X
+ XYX
[Ypq]
etc. as,
175
K
where,o
K
,
K
cX.,XYX
X =
Note that, C'.
etc.
t
f\,,
X + X YX =
(say) is a quadratic in
A
=o'.
X = (Xl,...,XN), similarly,
+ Xix and, we can
c\J
express,
y =
where S's are some coefficients.
A
Thus G() has a linear form in
and & and
.'s are quadratic in X's.
a. E. D.
Non-linear restrictions lead to PART-Il.
Proof:
PART-Il
From (I), we can write,
Yi
0'.
(J)
°j
i
Y
=
=
'jm
'j1
=
e
(say), i,j=1,..
.
,k.
'jN
pm
(K)
°'q
1q1
''qm
'qN
q
(say), p,q=k+1,.
..
,N
Consider (J) with i=1, j=j, then,
tia.
(L)
1
i.ii
ii
From (L), we get, y.
jm
=
e1
im
e3 (say)
11N
'1m
and iii =
for all j=1,.
. .
,k.
Now let m=i then above two results become
(M)
ji
ii
by substituting for
and
= Y
Y
=
11
'
1j1
= e'111.
Similarly from (K) we can show that,
= 0
(N)
1pq
where,
p,q = k+1,. .
for all p = k+1,.
.
.,N.
31
ii
i,j1,..
.
,k,
176
Consider (L) again, so that
jm = a3
Now limit m=p where p=k+1,.
(0)
=
a3
a3
m1,.
'1m
.
.
.,N
,N, then
.
from symmetry,
,
p1
11p=
pi
Consider (K), and allowing for q=N, we get,
= 1:21
N
=
=
1:P.i =
'N1
=
=
NN
Nl
from which we get,
p1
eR(Nl
=
by choosing 1 = 1 above.
,
From (0) and (P), we see that,
Y.
jp
(Q)
=O
I
= a3 e
p1
iN
(\J
Let us substitute (M), (N) and (a) into the vectors o and matrix I
So that
1k'
we get,
sk
(e'... k)
e'et.. e'ek
u
4
so,
=
eke
note that,
a a
=
0
,
where
a
(S... ek)
Also note that the cross-partials matrix,
F010 k+1
= [IJ
YiN)
ekek'1
can be written as,
=' e k+1
/
0N)
= ee, where, 6
(
a
N)
,c1
=0CM 0
and
II
k+1'"' °N
8k1
=
and I =
c'pq:J.
= 1PIRI
0, where
Similarly,
0k0N
177
The quadratic expression could be written as,
I
y = Ot.0+i
j
I
%Ij
+ x1(B X)
X
"I
X
I
+
A
AAA
I
X +oi X
X + X
+ X
X
(ee
(X
AA
(e X)
)X + o
+
A
X
NN
A Al
A
(Be )X
A
A1
+'lNX (ee)X
J.#
(e X)
+
(e
A1A
(0 x)
a
(e
x)
+
X)
+
NN
(a X)
(o x)
(0 x)
+ ''1N
This expression is recognized as
X)
quadratic in (0 X) and (0
under Non-linear Restrictions, we have, G(X1,..
.
).
Thus,
,X) expressed as a
quadratic in linear functions 0 X and 8 X.
Q.E.D.
Lemma:
If X1:lnX1 V 1, then OX is a Cobb-Douglas process func-
tion so that lnf = B X, etc. and writing the quadratic expansion of
lnY, in terms of these Cobb-Douglas forms, we get,
mY
= CL0 + CL1.
(a K)
(8 X) + y
.
(e'x)
AIA
+CLN.
(0 x)
NN
+
+ 11N
i.e. TRANSLOG:
my
+
lnf+
Y
(a x)
(e X)
(e x)
(a K).
AlA
(lnF)(ln)
I
(ln) (ln)
NN
+ CLN
+ '1N
(ln?)(ln).
Theorem 3A
If G(xl,...,xN) is quadratic in Xj'S and if G is weakly separable
with Nonlinear Separability Restrictions, then G is quadratic in
linear functions, f and g of
s and
s, respectively.
The Elasticity
178
of substitution between f and g is variable but finite, the elasticity
of substitution between X's or X's is however infinite.
-dfq g
Proof:
(ff + gth
g
)
(1')
g
fg
thff
g2 - 2
fgfg + ggg2
Note that, froi n Theorem 2 (PART II),
=ci
+31f + 3f2 + i1g + 1292 + Sfg
where, f,g are linear in Xs and Xs, re spectively.
Also,
+ 2f32f + 5g,
232
= y1 +2)g+ 5f,
2y2
gg
'fg
The terms within
for convexity of
.. .
brackets must be negative (i.e. non-zero)
- isoquandts in f-g space.
a finite function of f and g in general.
Therefore, we have,
Yfg
The linear f or g however
have infinite elasticities of substitution between X s or X s, respectively as shown in Theorem 38.
If G
x1,..., xN
is quadratic, then a linear WS restriction implies
that G is linear in quadratic process functions.
+cg(x) where a,b,c, are parameters and
separable groupds of process inputs.
,
x are vectors representing two
Thus linear WS restriction implies
that the elasticity of process-substitution
fg
Proof:
i.e. G(") = a+bf(x)
is infinitely large.
The elasticity of process-substitution is shown in (1),
where, G(")
c(f,g)
and f = f() and
g = g(x)
179
are process functions, and
is negative.
From Theorem 2 (PART-I)
we know that linear WS restriction gives rise to,
=
abf()
+
cg(),
so that
b, cbgcandsbff=cbgg
Thus,
fg
+ bc (bf + cg)
fg (0)
fg
=0.
+
Q.E.D.
Note:
The q-isoquandt in f-g space is straight lines with nega-
tive slopes as shown below contrasted with f and g- soquandts.
I
\
outut\
-isoquants
rocess_\'
f2
f-i soquants
f
q-isoauants
gi
x
p
APPENDIX B
Graph 1:
Capital Service K vs. Labor Service N
Eastern Oregon (Logarithms of $1000/acre)
K
-11.151.
-4.571
I
I
I
I
I
I
I
I
V
-5.01'.
.
V
-c.123
S
2
-5.233
I
S
2
I
1
I
V
S
.
I
V
V
-6.3113
I
-5.1153
I
I
S
I
I
I
V.
I
I
S
I
V
I
-6.TF.1
I
VII
V
-5.593 I
V.
I
II
V
I
I
I
S
V
S
VVVVIVIIIVSIVISISPVSIVIIIIVVSI IIIlIVIIIIlV SI.I..IIVVPI.111V1!IIVIIVI.VIII4IIN
V
S
I
I
S
-11.71.0
E.114
-5.195
I
S
1..252
I
I
-2.905
-3. 5?'.
-3.3f6
-2.1.51
0
APPENDIX B
Graph 2:
Capital Service K vs. Labor Service N
All Oregon (logarithms of $1000/acre)
K
-3.205
4
4
-3.555 4
*
4
.4
4
*
4
..
.
4
.4
.
4
4
v.
*
4
4
4
4
.
2.
?
.
4
.4
1
.
4
ft..
4
2 ..?
-53fl3
2
4
*1
4
4
4
.
t
-5.653
S
4
34
4
4
?
-6.003
*
* 4*444 44444 4 444 #4 44 4 4*4 8444* #4 8*4 4 4 *44 4 8* * 4 4 88 * * 4* * * 4444 48 4 *48* * 4
*
4
-6.526
4
4
-5.271+
-5.933
4
4
-3. 955
-4.6t'
4
-2.637
-3.296
-1
co
a
182
APPENDIX C
List of Weather Stations Chosen
County
Station
Station
County
Baker
Baker CAA Airport
Linn
Albany
Benton
Corvallis
Malheur
Malheur Br. Exp. Stn.
Clackamas
Canby
Marion
Salem
Clatsop
Astoria AP
Multnomah
Bonneville Dam
Coos
North Bend
Polk
Dallas
Crook
Prineville
Sherman
Kent
Curry
Brookings
Tillamook
Tillamook
Deschutes
Redmond
Umatilla
Pendletori
Douglas
Roseburg
Union
LeGrande AP
Gilliam
Arlington
Wallowa
Wallowa
Grant
John Day
Wasco
The Dalles
Harney
Burns WB City
Washington
Canary
Hood River
Pardale
Wheeler
Fosill
Jackson
Medford Exp. Station
Yamhill
McMinville
Jefferson
Madras
Josephine
Grants Pass
Klamath
Kiamath Falls AP
Lake
Paisley
Lane
Eugene
Lincoln
Newport
183
Data Summary of Log-transformed Variables
Eastern Oregon
a)
Variables
Means
Standard Deviations
Y
3.58
0.43
I
-2.96
2.10
F
-2.54
0.91
L
-3.85
0.71
K
-4.67
0.41
p
5.18
0.19
T
3.85
0.46
Correlation Matrix
V
Y
I
1.00
-0.14
1.00
I
F
L
K
P
T
symmetrical
F
T
L
K
0.48
-0.19
-0.14
0.28
0.11
-0.32
-0.52
-0.42
-0.80
-0.45
1.00
-0.11
-0.11
0.21
0.18
1.00
0,74
0.08
0.08
1.00
0.06
0.20
1.00
0.04
P
1.00
184
APPENDIX D (Contd.)
Data Summary of Log-transformed Variables
b) Western Oregon
Variables
Means
Standard Deviations
Y
3.50
0.34
I
-2.68
2.71
F
-3.46
1.12
L
-4.93
0.72
K
-5.26
0.43
P
2.44
0.45
Correlation Matrix
V
V
I
F
L
K
1.00
0.28
0.76
0.24
0.50
0.12
1.00
0.45
0.41
0.59
-0.02
1.00
0.27
0.62
-0.01
1.00
0.81
0.32
1.00
0.26
I
F
L
K
P
Symmetrical
P
1.00
APPENDIX E
T.S.P. Output
TIME SERIES PROCESSOR D.P. VERSiON 3.3
EQUAIION
OCT,1927
CYDER 73
81/04/2?.
15.35.34.
RUN55
1
* 4 * 44444 * * 4*
ORDINARY LEAST SQUARES
DEPENDENT VARIABLE:
RIGHTHAND
VARIABLE
P
F
1
K
1(1
C
Y
ESTIMATED
COEFFICIENT
.136729
.196526
.589921
.269431
.105930
5.27635
STANDARD
ERROR
SIAIISTIC
.51143?E-01
.256568E-01
.124186
.8?3099E-01
.21??84E-01
.449438
2.67344
7.661??
4.75031
3.08591
4.86398
11.2399
LOG OF LIKELIHOOD FUNCTION
21.1844
R-SOUARED
.68??
OURBIN-UATSON STATISTIC (ADJ. FOR 4. GAPS)
1.7481
SUM OF SQUARED RESIDUALS
3.18355
STANDARD ERROR OF THE REGRESSION
.197038
SUM OF RESIDUALS
.343903E-11
NUMBER OF OBSERVATIONS
88.
MEAN OF DEPENDENT VARIABLE
3.49789
F-STAIIST{C(
82.)
36.1 178
5.,
-4
Co
(51
APPENDIX E
(Continued)
ESTIhATE OF VARIANCE-COVAIAtWE MAIRIX OF EST1;TE
P
F
I
CUEFF:LCIENTS
C
1<1
.261 568E-02
. 15251 2E-03
.1 40?4?E-O'2
- 904958E-03
.230927E-O:3
-.1 0260?E-01
F
.152512E-03
.658269E-03
-.966841E-03
I
.1 40?47E-02
-.966841 E--03
. 154221 E-01
-.904958E--03
-.12?941E-02
-.165693E-03
-.496513E-02
.53696?E-02
.269605E-02
.229432E-01
-.127941E-02
.536967E-02
.762302E-02
.100511E-02
.3?4041E-01
-.165693E-03
269605E-02
.100511E-02
.4?4300E-03
.436429E-02
-.496513E-02
229432E-01
.3?4041E-01
.436429E-02
.201995
F
K
.
.
.230927E-03
C
-. 102607E-01
1
2
3
4
5
6
187
APPENDIX F
Data Set in Logarithms of Variables
The variables are
The row with five
Y, I, F, N and K,
The next row with
presents the corn
respectively.
given as follows.
elements represents
respectively.
two elements responding T and P,
i1.7a5t
-.ô3.77
-
-.537á.3
.Z,179
367
-5..3i.35áó
-3.Q292Q3h
Z.723i.73
3,j75j
IS2r
.3172TA
.33325Z7
-t.05527â3
.à3'729
-L.'+9it
3.ó6ó2S
-.7833ó
-.7gQa3i
L.9+91i.
-.53259
.3522â
*.52.3C3
-t.tô725
-1.t33ii
-..àii1Tâ
-5.3TS432
-5.ôi9O33
4.5.95ó7
7.O5ifl.2Z5
-1.25722
5.137à
2.8,ó53Z
-#.235c26
-1.USá237
2.4á49Já6
-.3i.2U29
Z.3325t
.3â135
Z.ó+'âô
-ii...lTh
-3.335
-.ó2797
-.32T35
-.5iâ5
2. 3C 28 51
.i73.84+
.55373
2.32351.
2.3721c5
2.7323
2 53 9C 573
-1.2l.975
-. o-31232
i..'2S23QÔ
-..273227i.
2.53Q373
-3.
93
2.'849ãã
-D.o2àê8
-237S53o%
5.L1â]à2
.L3.81
-'..7233
.23.5C17
-3.ZU32Y
5.35526
5.17O'8'3
3.737oS6
_______3.3i.25
-i.ZC9225
-5.U2T333_Ti2fl3
-3.25oT2
-3.37n:3
Data Set in Logarithnis of 'Jariables (continued)
-L.Qããt23
5.552C.
-i.53áL3Q
-i..a8123
-1,Ci23
-..)21275
-3.5i.â777
-..32à+335
-.375917
-3..312313
2.7.o97
.37ã+L7
.937e5*
-3.737G427
-3.1643i
-...73i52
-5.,1579g
-3.3Q.3ä7.
-3.ô7.
-.5299C2
-..i.L-39
4.i.431347
-3.5351;
3..J97
-1.339..]
'..3b735
-..Si1e74Q
-.82337
3.s57
-...37ó759
2.39J.37i
3.à3?32
-1.!53ê3
3. 32
-1.3o3Q2
-3.U2'.5ä13
3.7612Ui
-i..3âo227
5.3222
i.,t3','3
t.951ii.7
2.5,233
-3.2252
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Data Set in Logarithms of Variables (continued)
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192
Data Set in Logarithms of Variables (continued)
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